#8696732

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

4204 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=g,e^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, 8603928);