#455895

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1154 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:

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

Polycyclic presentation:

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

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, 363091);