#471620

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1300 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378816);