#508538

order	512
nilpotency class	4
isoclinism	isoclinic class of #508494
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

487 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,26)(13,18)(14,32)(15,20)(17,28)(19,31)(21,38)(22,39)(23,55)(24,42)(25,40)(27,59)(29,44)(30,50)(33,65)(34,47)(35,67)(36,49)(37,45)(41,57)(43,62)(46,64)(48,51)(52,66)(53,69)(54,68)(56,72)(58,71)(60,63)(61,70);(2,5)(6,8)(9,11,21,24,13,25,15,36)(10,16,38,42,18,40,20,49)(12,27,37,61,28,62,30,56)(14,33,51,57,31,64,35,23)(17,43,50,72,26,59,45,70)(19,46,67,55,32,65,48,41)(22,52,53,29,54,34,58,63)(39,66,69,44,68,47,71,60);(3,7)(6,8)(9,12,13,28)(10,17,18,26)(11,22,25,54)(14,34,31,52)(15,37,21,30)(16,39,40,68)(19,47,32,66)(20,50,38,45)(23,56,57,61)(24,58,36,53)(27,33,62,64)(29,35,63,51)(41,70,55,72)(42,71,49,69)(43,46,59,65)(44,48,60,67);(1,3,4,7)(2,6,5,8)(10,18)(16,40)(17,26)(19,32)(20,38)(39,68)(41,55)(42,49)(43,59)(44,60)(45,50)(46,65)(47,66)(48,67)(69,71)(70,72)

Polycyclic presentation:

2^9|b^2=fh,c^2=i,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=de,g^c=ghi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415734);