#6372829

order	512
nilpotency class	3
isoclinism	isoclinic class of #60372
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2701 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280025);