#7043942

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#45412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1508 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6951138);