#372328

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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×C₂²⋊C₄
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1175 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,25,29,37)(13,30)(14,32)(15,20)(16,21)(17,24)(18,28,38,33)(19,31)(23,36,27,34)(26,40,39,35);(9,11)(10,17)(12,26,29,39)(13,16)(14,31)(15,24)(18,35,38,40)(19,21)(20,30)(22,32)(23,28,27,33)(25,36,37,34);(2,5)(6,8)(9,12)(10,18)(11,23)(13,27)(14,28)(15,33)(16,29)(17,34)(19,36)(20,37)(21,38)(22,35)(24,39)(25,32)(26,31)(30,40);(1,3,4,7)(2,6,5,8)(9,13,16,11)(10,19,21,17)(12,27,29,23)(14,24,15,31)(18,36,38,34)(20,22,32,30)(25,40,37,35)(26,28,39,33)

Polycyclic presentation:

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

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, 279524);