#7011707

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	#28233	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1415 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,27,22,28,21,38)(19,29,33,46,32,24,31,42)(20,34,36,44,37,26,39,25)(30,41,47,40,35,45,48,43);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,32)(18,24,28,29)(20,35,37,30)(21,31,23,33)(25,43,44,40)(26,45,34,41)(27,42,38,46)(36,48,39,47);(9,11)(10,13)(12,16)(14,15)(17,20,22,37)(18,25,28,44)(19,30,32,35)(21,36,23,39)(24,40,29,43)(26,38,34,27)(31,47,33,48)(41,46,45,42);(2,5)(6,8)(11,16)(13,15)(17,20)(18,26)(19,30)(21,39)(22,37)(23,36)(24,41)(25,27)(28,34)(29,45)(31,48)(32,35)(33,47)(38,44)(40,42)(43,46);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,23)(18,27,28,38)(19,31,32,33)(20,36,37,39)(24,42,29,46)(25,26,44,34)(30,47,35,48)(40,41,43,45)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,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, 6918903);