#378801

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

762 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)(3,6)(4,5)(7,8)(9,10,15,20,16,21,14,31)(11,17,13,19,24,33,23,38)(12,25,28,18,30,36,29,37)(22,34,27,32,39,35,26,40);(2,5)(6,8)(9,11,15,13,16,24,14,23)(10,17,20,19,21,33,31,38)(12,26,29,39,30,27,28,22)(18,34,25,40,37,35,36,32);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,22)(13,27)(14,29)(15,28)(16,30)(17,32)(19,35)(20,36)(21,37)(23,26)(24,39)(25,31)(33,40)(34,38);(2,5)(6,8)(9,13,15,24,16,23,14,11)(10,19,20,33,21,38,31,17)(12,27,28,39,30,26,29,22)(18,35,36,40,37,34,25,32)

Polycyclic presentation:

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

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, 285997);