#8696312

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4166 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=gh,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, 8603508);