#372336

order	512
nilpotency class	3
isoclinism	isoclinic class of #222240
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=fgi,e^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,e^b=ei,f^b=fi,g^b=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, 279532);