#330592

order	512
nilpotency class	3
isoclinism	isoclinic class of #329587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

595 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)(9,10,13,16,12,15,11,14)(17,18,20,22)(19,25,27,23)(21,24,28,30)(26,31,29,32);(2,5)(6,8)(18,22)(19,26)(21,28)(23,31)(25,32)(27,29);(3,7)(6,8)(9,10,12,15)(11,16,13,14)(17,19)(18,23)(20,27)(21,29)(22,25)(24,31)(26,28)(30,32);(1,3,4,7)(2,6,5,8)(18,22)(19,27)(24,30)(26,29)

Polycyclic presentation:

2^9|a^2=efhi,c^2=h,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, 237788);