#7609719

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1759 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,36)(28,33)(29,37)(30,39)(31,34)(32,35)(38,43)(40,47)(41,44)(42,45)(46,48);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,23)(20,30)(22,32)(24,33)(25,38)(27,40)(29,42)(31,43)(34,39)(36,46)(41,48)(44,47);(10,14)(11,15)(17,19)(18,24)(20,27)(21,28)(22,29)(23,33)(25,36)(26,37)(30,40)(31,41)(32,42)(34,44)(35,45)(38,46)(39,47)(43,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,22,30,21,31,32,43)(18,25,26,39,23,34,35,38)(19,27,29,40,28,41,42,48)(24,36,37,47,33,44,45,46);(2,7)(6,8)(18,23)(19,28)(25,34)(26,35)(27,41)(29,42)(38,39)(40,48)

Polycyclic presentation:

2^9|b^2=fi,d^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7516915);