#363052

order	512
nilpotency class	3
isoclinism	isoclinic class of #188176
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

1094 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270248);