#378096

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

861 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)(3,8)(4,6)(5,7)(11,19,22,25)(15,27,20,31)(21,33,35,36)(28,38,34,39);(1,3,5,4)(2,6,7,8)(9,10,13,17,14,18,26,32)(11,20,19,31,22,15,25,27)(12,16,23,29,24,30,37,40)(21,34,33,39,35,28,36,38);(1,4,5,3)(2,6,7,8)(9,11)(10,15)(12,21)(13,25)(14,22)(16,28)(17,31)(18,20)(19,26)(23,36)(24,35)(27,32)(29,39)(30,34)(33,37)(38,40);(2,7)(6,8)(9,12,13,23,14,24,26,37)(10,16,17,29,18,30,32,40)(11,21,25,36,22,35,19,33)(15,28,31,39,20,34,27,38)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285292);