#373800

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1118 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)(3,7)(4,5)(6,8)(9,10,13,18,14,17,24,31)(11,15,21,27,20,26,34,38)(12,16,23,30,22,29,35,40)(19,28,33,39,32,25,36,37);(2,5)(6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,28)(17,26)(18,27)(22,32)(23,33)(24,34)(25,29)(30,39)(31,38)(35,36)(37,40);(9,12,13,23,14,22,24,35)(10,16,18,30,17,29,31,40)(11,19,21,33,20,32,34,36)(15,25,27,37,26,28,38,39);(1,3,4,8)(2,6,5,7)(12,22)(16,29)(19,32)(23,35)(25,28)(30,40)(33,36)(37,39)

Polycyclic presentation:

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

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, 280996);