#455681

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23390	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1163 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,23)(21,25)(22,26)(24,30)(27,32)(29,31);(1,2,4,5)(3,6,7,8)(9,11,12,15)(10,13,14,16)(17,18,21,25)(19,29,20,24)(22,27,32,26)(23,31,28,30);(1,3)(2,6)(4,7)(5,8)(10,14)(13,16)(17,19,21,20)(18,23,25,28)(22,31,32,30)(24,27,29,26);(3,7)(6,8)(17,20)(18,24)(19,21)(22,31)(23,26)(25,29)(27,28)(30,32)

Polycyclic presentation:

2^9|b^2=efi,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,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, 362877);