#6815022

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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

2024 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)(13,14,18,22)(15,24,27,19)(16,23,28,20)(17,21,25,26);(5,6)(7,10)(8,9)(11,12)(13,14)(15,25)(16,20)(17,27)(18,22)(19,26)(21,24)(23,28);(1,2,3,4)(13,15,16,26,18,27,28,21)(14,19,23,25,22,24,20,17);(5,7,8,11)(6,10,9,12)(13,16,18,28)(14,20,22,23)(15,26,27,21)(17,24,25,19);(6,9)(10,12)(13,17,28,24,18,25,16,19)(14,21,20,27,22,26,23,15)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=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, 6722218);