#323634

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230830);