#256525

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

779 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,29)(21,26)(22,27)(23,28)(25,31)(30,32);(1,3,5,4)(2,6,7,8)(10,13)(12,16)(17,19,23,29)(18,24,28,20)(21,25,22,32)(26,31,27,30);(1,4,5,3)(2,6,7,8)(10,14)(12,16)(13,15)(17,20,23,24)(18,25,28,32)(19,26,29,27)(21,30,22,31);(2,7)(6,8)(10,13)(14,15)(17,21,23,22)(18,26,28,27)(19,25,29,32)(20,30,24,31)

Polycyclic presentation:

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

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, 163721);