#7614461

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1916 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6)(7,11)(8,12)(9,10)(13,14)(15,25)(16,21)(17,20)(18,23)(19,24)(22,27)(26,28);(2,4)(6,10)(11,12)(13,15)(14,21)(16,18)(17,22)(19,27)(20,26)(23,25)(24,28);(5,7,9,8)(6,11,10,12)(13,16,19,27,18,15,20,26)(14,22,17,21,23,28,24,25);(1,2,3,4)(13,17,20,14,18,24,19,23)(15,21,27,22,16,25,26,28);(5,8,9,7)(6,11,10,12)(14,23)(17,24)(21,25)(22,28)

Polycyclic presentation:

2^9|c^2=fgi,d^2=h,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,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, 7521657);