#256514

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,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, 163710);