#7162178

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1533 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,22,26)(19,27,30,43)(20,31,33,24)(21,35,36,25)(23,37,39,29)(28,44,32,42)(34,41,45,48)(38,47,40,46);(2,5)(3,7)(10,13)(11,15)(17,19,21,29,22,30,36,37)(18,23,25,27,26,39,35,43)(20,32,34,46,33,28,45,47)(24,40,41,44,31,38,48,42);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,28)(21,34)(22,33)(23,38)(25,41)(26,31)(27,42)(29,47)(30,32)(35,48)(36,45)(37,46)(39,40)(43,44);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,29)(20,33)(21,36)(23,27)(24,31)(25,35)(28,46)(30,37)(32,47)(38,44)(39,43)(40,42);(9,11,12,15)(10,14,13,16)(19,30)(20,33)(23,39)(24,31)(27,43)(29,37)(34,45)(41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

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, 7069374);