#465933

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ 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:

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

Polycyclic presentation:

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

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, 373129);