#6810258

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,19,26,25)(16,20,30,28)(17,21,32,36)(23,33,38,27)(24,34,39,31)(29,35,43,41)(37,44,42,40);(5,7,8,12)(6,9,10,11)(13,15,18,26)(14,19,22,25)(16,27,30,33)(17,31,32,34)(20,23,28,38)(21,24,36,39)(29,42,43,37)(35,40,41,44);(2,4)(7,12)(9,11)(13,16,22,28,18,30,14,20)(15,23,19,33,26,38,25,27)(17,29,36,41,32,43,21,35)(24,37,34,44,39,42,31,40);(6,10)(9,11)(13,14,18,22)(15,19,26,25)(16,28,30,20)(17,21,32,36)(23,27,38,33)(24,34,39,31)(29,41,43,35)(37,40,42,44);(13,17,22,36,18,32,14,21)(15,24,19,34,26,39,25,31)(16,29,28,41,30,43,20,35)(23,37,33,44,38,42,27,40)

Polycyclic presentation:

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

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, 6717454);