#7162181

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	#30995	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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)(17,18,22,25)(19,27,30,41)(20,31,32,24)(21,34,35,26)(23,36,38,29)(28,42,44,40)(33,39,47,46)(37,43,45,48);(2,5)(3,7)(10,13)(11,15)(17,19,21,29,22,30,35,36)(18,23,26,27,25,38,34,41)(20,28,33,43,32,44,47,48)(24,37,39,40,31,45,46,42);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,32)(18,24,25,31)(19,28,30,44)(21,33,35,47)(23,37,38,45)(26,39,34,46)(27,40,41,42)(29,43,36,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,29)(21,35)(23,27)(26,34)(28,43)(30,36)(33,47)(37,40)(38,41)(39,46)(42,45)(44,48);(9,11,12,15)(10,14,13,16)(18,25)(20,32)(23,38)(26,34)(27,41)(28,44)(33,47)(43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=i,d^2=gi,e^2=h,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, 7069377);