#6814917

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1965 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)(5,6)(7,11)(8,10)(9,12)(13,14)(15,25)(16,27)(17,24)(18,22)(19,23)(20,26)(21,28);(5,7)(6,9)(8,12)(10,11)(13,15,19,26)(14,20,23,25)(16,28,22,17)(18,21,27,24);(1,2,3,4)(13,16,17,15,19,22,28,26)(14,18,24,20,23,27,21,25);(6,10)(9,11)(13,17,19,28)(14,21,23,24)(15,16,26,22)(18,20,27,25);(7,12)(9,11)(13,18,17,20,19,27,28,25)(14,22,24,26,23,16,21,15)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722113);