#382092

order	512
nilpotency class	3
isoclinism	isoclinic class of #219508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

907 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)(4,5)(6,8)(9,10,13,18,14,19,27,15)(11,16,22,30,23,32,21,28)(12,17,25,34,26,35,38,29)(20,31,37,40,36,39,24,33);(2,5)(6,7)(10,15)(13,27)(16,30)(17,29)(18,19)(21,22)(24,37)(25,38)(28,32)(31,40)(33,39)(34,35);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,24)(13,22)(14,23)(15,28)(17,33)(18,30)(19,32)(20,25)(21,27)(26,37)(29,39)(31,34)(35,40)(36,38);(3,8)(6,7)(9,12,14,26)(10,17,19,35)(11,20)(13,25,27,38)(15,29,18,34)(16,31)(21,24)(22,37)(23,36)(28,33)(30,40)(32,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289288);