#7035550

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1346 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)(11,13)(12,14)(15,16)(17,18,24,30)(19,25,35,43)(20,36,21,27)(22,29,23,28)(26,38,44,37)(31,45,32,40)(33,42,34,41)(39,47,48,46);(1,3,5,8)(2,6,4,7)(9,11,13,10)(12,14,16,15)(17,19,24,35)(18,25,30,43)(20,31,21,32)(22,33,23,34)(26,39,44,48)(27,40,36,45)(28,41,29,42)(37,46,38,47);(10,11)(12,16)(17,20)(18,26)(19,31)(21,24)(22,37)(23,38)(25,39)(27,29)(28,36)(30,44)(32,35)(33,46)(34,47)(40,42)(41,45)(43,48);(1,4,5,2)(3,6,8,7)(9,12,13,16)(10,14,11,15)(17,21,24,20)(18,27,30,36)(19,32,35,31)(22,37,23,38)(25,40,43,45)(26,28,44,29)(33,46,34,47)(39,41,48,42);(10,11)(14,15)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,37,21,38)(25,41,43,42)(26,36,44,27)(31,46,32,47)(39,45,48,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei

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, 6942746);