#511653

order	512
nilpotency class	4
isoclinism	isoclinic class of #511577
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

523 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18,21,26)(19,28,31,24)(20,32,23,25)(22,27,34,42)(29,44,45,38)(30,39,37,40)(33,47,35,41)(36,48,46,43);(1,3)(2,5)(4,7)(6,8)(9,11,13,16)(10,14,12,15)(19,29)(20,23)(22,35)(24,38)(25,32)(27,41)(28,44)(30,46)(31,45)(33,34)(36,37)(39,43)(40,48)(42,47);(2,6)(5,8)(10,12)(14,15)(17,19,21,31)(18,24,26,28)(20,30,23,37)(22,36,34,46)(25,39,32,40)(27,43,42,48)(29,33,45,35)(38,47,44,41);(3,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20,21,23)(18,25,26,32)(19,30,31,37)(22,33,34,35)(24,39,28,40)(27,41,42,47)(29,36,45,46)(38,48,44,43)

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=i,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418849);