#7592211

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#45236	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1618 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)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,34)(21,39)(22,28)(23,29)(24,30)(26,37)(27,33)(31,42)(32,41)(35,40)(36,45)(38,44)(43,46)(47,48);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,35)(21,22)(23,32)(24,33)(26,42)(27,30)(28,39)(29,41)(31,37)(34,40)(36,47)(38,46)(43,44)(45,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,23,37,24,38,22,36)(18,26,28,34,30,45,29,44)(19,31,21,35,33,47,32,46)(25,40,41,42,27,43,39,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21,23,33,24,32,22,19)(18,27,28,39,30,25,29,41)(20,35,37,47,38,46,36,31)(26,43,34,48,45,40,44,42);(2,6)(7,8)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,21,33,32)(20,36,38,37)(25,39,27,41)(26,34,45,44)(31,35,47,46)(40,48,43,42)

Polycyclic presentation:

2^9|a^2=gh,c^2=fg,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7499407);