#7582411

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#25538	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1548 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7489607);