#8696259

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	#49772	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=h,e^2=h,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, 8603455);