#8696001

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	#37038	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 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)(3,12,6,8)(4,14,15,9)(5,10,16,13)(17,18,24,23)(19,28,22,25)(20,27,21,26)(29,31,30,32)(33,34,39,43)(35,40,44,46)(36,41,45,48)(37,38,42,47);(1,3,7,6)(2,8,11,12)(4,10,15,13)(5,9,16,14)(17,19,24,22)(18,25,23,28)(20,30,21,29)(26,32,27,31)(35,44)(36,45)(40,46)(41,48);(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,35,39,44)(34,40,43,46)(36,38,45,47)(37,41,42,48);(1,4,7,15)(2,9,11,14)(3,13,6,10)(5,12,16,8)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,36)(34,41)(35,38)(37,46)(39,45)(40,42)(43,48)(44,47);(1,5)(2,10)(3,9)(4,12)(6,14)(7,16)(8,15)(11,13)(17,22,24,19)(18,28,23,25)(20,30,21,29)(26,32,27,31)(33,37)(34,38)(35,41)(36,46)(39,42)(40,45)(43,47)(44,48);(1,6,7,3)(2,8,11,12)(4,13,15,10)(5,9,16,14)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(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=g,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, 8603197);