#450831

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1032 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358027);