#387105

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=gi,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,h^c=hi,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, 294301);