#7144330

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#45248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1760 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7051526);