#7145794

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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

2223 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=f,e^2=hi,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=dhi,e^c=ei

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, 7052990);