#370541

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1237 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)(4,8)(5,7)(9,10)(11,20)(12,25,29,36)(13,18)(14,19)(15,22)(16,28,34,27)(17,30)(21,31,38,40)(23,32)(24,35)(26,33,39,37);(1,3,5,4)(2,6,7,8)(9,11,14,24)(10,15,19,32)(12,26)(13,23,30,22)(16,33)(17,35,18,20)(21,27)(25,40)(28,38)(29,39)(31,36)(34,37);(1,4,5,3)(2,6,7,8)(9,12,14,29)(10,16,19,34)(11,21,24,38)(13,27,30,28)(15,31,32,40)(17,25,18,36)(20,37,35,33)(22,26,23,39);(9,11)(10,15)(12,21)(13,23)(14,24)(16,31)(17,35)(18,20)(19,32)(22,30)(25,33)(26,28)(27,39)(29,38)(34,40)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^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, 277737);