#6815005

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2071 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)(6,9)(7,11)(13,14)(15,23)(16,25)(17,28)(18,20)(19,22)(21,26)(24,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,19)(18,25)(20,23)(21,22)(24,28)(26,27);(1,2,3,4)(13,16,27,21,18,23,19,28)(14,15,24,17,20,25,22,26);(5,7,8,11)(6,10,9,12)(14,20)(16,23)(21,28)(22,24);(6,9)(10,12)(13,17,19,15,18,26,27,25)(14,21,22,16,20,28,24,23)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722201);