#7567202

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1562 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)(18,23)(19,27)(20,31)(26,36)(30,43)(34,37)(39,41)(45,48);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,23)(19,28,27,24)(20,32,31,42)(22,36,35,26)(25,37,38,34)(29,45,33,48)(30,40,43,44)(39,46,41,47);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,24)(20,33)(21,27)(22,30)(23,28)(25,41)(26,40)(29,31)(32,45)(34,47)(35,43)(36,44)(37,46)(38,39)(42,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20)(18,25)(19,29)(21,31)(22,37)(23,38)(24,39)(26,42)(27,33)(28,41)(30,47)(32,36)(34,35)(40,45)(43,46)(44,48);(2,6)(7,8)(10,14)(13,15)(19,27)(20,31)(24,28)(25,38)(30,43)(32,42)(34,37)(40,44)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7474398);