#7544122

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1555 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fg,e^2=i,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,e^d=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, 7451318);