#377148

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1529 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284344);