#382104

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289300);