#6995878

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1534 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=i,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=ehi,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6903074);