#7027402

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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1599 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)(11,13)(12,14)(15,16)(17,18,22,27)(19,28,23,24)(20,31,33,38)(21,26,37,44)(25,29,41,34)(30,42,35,39)(32,47,36,48)(40,45,43,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,28)(20,29,33,34)(21,35,37,30)(25,38,41,31)(26,42,44,39)(32,46,36,45)(40,47,43,48);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,19,29,22,33,23,34)(18,25,24,38,27,41,28,31)(21,36,30,46,37,32,35,45)(26,43,39,47,44,40,42,48);(2,6)(7,8)(11,16)(13,15)(17,21,22,37)(18,26,27,44)(19,30,23,35)(20,32,33,36)(24,39,28,42)(25,40,41,43)(29,45,34,46)(31,47,38,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(20,33)(21,37)(25,41)(26,44)(29,34)(30,35)(31,38)(39,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=i,e^2=fi,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, 6934598);