#373351

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10,14,21,16,22,15,20)(11,23,25,17,26,32,30,33)(12,19,28,36,29,35,13,18)(24,37,27,34,38,31,39,40);(1,3,5,4)(2,6,8,7)(9,11,16,26)(10,17)(12,27,29,39)(13,24,28,38)(14,25,15,30)(18,34)(19,31)(20,23)(21,32)(22,33)(35,37)(36,40);(1,4,5,3)(2,7,8,6)(9,12,15,13,16,29,14,28)(10,18,21,19,22,36,20,35)(11,24,25,27,26,38,30,39)(17,31,23,34,33,37,32,40);(2,8)(6,7)(9,13,15,29,16,28,14,12)(10,19,21,36,22,35,20,18)(11,24,30,39,26,38,25,27)(17,31,32,40,33,37,23,34)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,d^2=efg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di

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, 280547);