#455095

order	512
nilpotency class	3
isoclinism	isoclinic class of #454981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2619
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16260
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1483 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 362291);