#7135178

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1465 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=hi,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,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, 7042374);