#8696675

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#54977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603871);