#7145809

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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

2167 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)(7,11)(10,12)(13,14,17,23,18,22,32,33)(15,20,26,35,25,37,31,19)(16,27,29,42,30,21,38,39)(24,40,28,34,41,36,43,44);(6,9)(7,11)(13,15,18,25)(14,19,22,35)(16,28,30,43)(17,31,32,26)(20,23,37,33)(21,36,27,40)(24,38,41,29)(34,42,44,39);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,24)(17,26)(18,25)(19,33)(21,36)(22,37)(23,35)(27,40)(28,29)(30,41)(31,32)(34,42)(38,43)(39,44);(6,9)(10,12)(13,16,17,29,18,30,32,38)(14,21,23,39,22,27,33,42)(15,24,26,28,25,41,31,43)(19,34,20,36,35,44,37,40);(5,7,8,11)(6,10,9,12)(14,22)(15,25)(21,27)(23,33)(24,41)(26,31)(28,43)(39,42)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7053005);