#7174987

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#45267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1670 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)(9,10,14,11)(12,15,13,16)(17,18)(19,23)(20,31)(21,25)(22,27)(24,28)(26,30)(29,39)(32,47)(33,41)(34,38)(35,48)(36,43)(37,44)(40,45)(42,46);(1,2)(3,7)(4,8)(5,6)(10,11)(12,13)(17,19,22,30)(18,23,27,26)(20,28,34,33)(21,29,37,36)(24,38,41,31)(25,39,44,43)(32,45,35,46)(40,48,42,47);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,15,13,16)(17,20,30,33,22,34,19,28)(18,24,23,38,27,41,26,31)(21,35,36,45,37,32,29,46)(25,42,39,47,44,40,43,48);(1,4,5,3)(2,7,6,8)(9,12,14,13)(10,15,11,16)(17,21)(18,25)(19,29)(20,32)(22,37)(23,39)(24,40)(26,43)(27,44)(28,45)(30,36)(31,47)(33,46)(34,35)(38,48)(41,42);(9,13,14,12)(10,15,11,16)(17,19,22,30)(18,23,27,26)(20,33,34,28)(21,36,37,29)(24,31,41,38)(25,43,44,39)(32,45,35,46)(40,48,42,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgi,d^2=f,e^2=hi,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^c=ei,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, 7082183);