#246964

order	512
nilpotency class	3
isoclinism	isoclinic class of #246957
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1011 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1011 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154160);