#493940

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1096 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)(9,10,12,17,13,18,23,30,14,16,22,28,24,29,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40);(2,5)(6,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,25)(17,26)(18,27)(22,31)(23,33)(24,32)(28,35)(29,36)(30,37)(34,39)(38,40);(10,16)(11,19)(12,22)(18,29)(21,32)(23,34)(26,35)(37,40);(1,3,4,7)(2,6,5,8)(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40)

Polycyclic presentation:

2^9|a^2=g,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=di,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401136);