#7195122

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	#45248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1710 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)(10,13)(11,15)(17,18,22,27)(19,28,33,43)(20,24,37,42)(21,25,29,26)(23,31,39,32)(30,45,34,48)(35,41,36,44)(38,47,40,46);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,13)(15,16)(17,19,29,31,22,33,21,32)(18,23,25,43,27,39,26,28)(20,34,35,46,37,30,36,47)(24,40,44,45,42,38,41,48);(2,5)(6,8)(10,13)(14,16)(17,20)(18,24)(19,30)(21,36)(22,37)(23,38)(25,44)(26,41)(27,42)(28,45)(29,35)(31,47)(32,46)(33,34)(39,40)(43,48);(9,11,12,15)(10,14,13,16)(17,21,22,29)(18,25,27,26)(19,31,33,32)(20,35,37,36)(23,28,39,43)(24,41,42,44)(30,46,34,47)(38,48,40,45)

Polycyclic presentation:

2^9|a^2=i,c^2=fghi,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, 7102318);