#7519011

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(5,6)(7,8)(9,10,14,16)(11,17,21,24)(12,20,22,15)(13,23,19,18);(1,2)(3,4)(5,6)(7,8)(9,11,19,12,14,21,13,22)(10,15,23,24,16,20,18,17);(6,8)(9,10)(11,20)(12,17)(13,18)(14,16)(15,21)(19,23)(22,24);(2,4)(6,8)(10,16)(11,21)(12,22)(18,23);(9,12,13,11,14,22,19,21)(10,17,18,20,16,24,23,15)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426207);