#8696244

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3865 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,d^2=g,e^2=gh,f^2=g,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, 8603440);