#7154899

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

2208 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7062095);