#7195664

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

2448 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,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, 7102860);