#7575836

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1547 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)(3,7)(4,8)(5,6)(10,14)(11,12)(17,18)(19,23)(20,24)(21,26)(22,25)(27,34)(28,40)(29,35)(30,36)(31,37)(32,38)(33,39)(41,48)(42,46)(43,45)(44,47);(2,6)(7,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,30)(18,23,25,36)(20,27,31,42)(21,28,33,29)(24,34,37,46)(26,40,39,35)(32,41,44,43)(38,48,47,45);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,27)(21,32)(22,31)(23,34)(25,37)(26,38)(28,41)(29,43)(30,42)(33,44)(35,45)(36,46)(39,47)(40,48);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,16,14,15)(18,25)(19,28)(21,33)(23,35)(24,37)(27,41)(29,30)(32,44)(34,45)(36,40)(42,43)(46,48);(2,6)(7,8)(9,12,13,11)(10,15,14,16)(20,31)(24,37)(27,42)(32,44)(34,46)(38,47)(41,43)(45,48)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7483032);