#408199

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

971 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)(3,8)(4,6)(5,7)(9,10,15,19)(11,20,23,33)(12,17,29,36)(13,30,18,14)(16,24,32,22)(21,34,39,37)(25,31,38,40)(26,28,35,27);(1,3,5,4)(2,6,7,8)(9,11,15,23)(10,16,19,32)(12,25,29,38)(13,20,18,33)(14,24,30,22)(17,34,36,37)(21,27,39,28)(26,31,35,40);(9,12)(10,17)(11,21)(13,26)(14,27)(15,29)(16,31)(18,35)(19,36)(20,37)(22,38)(23,39)(24,25)(28,30)(32,40)(33,34);(1,4,5,3)(2,6,7,8)(9,13,14,10,15,18,30,19)(11,20,24,16,23,33,22,32)(12,26,27,17,29,35,28,36)(21,37,25,31,39,34,38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=efh,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=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, 315395);