#6373780

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280976);