#7551112

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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
prime radical	C₂×#3275	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2090 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,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, 7458308);