#7623907

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#33579	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1282 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi

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, 7531103);