#6570630

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2181 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gi,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=eh

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, 6477826);