#8696280

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

4213 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,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, 8603476);