#330596

order	512
nilpotency class	3
isoclinism	isoclinic class of #329587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

630 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=i,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237792);