#8137193

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

4264 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)(3,12)(4,5)(6,14)(7,11)(8,16)(9,15)(10,13)(17,18,22,26)(19,23,24,20)(21,29,30,25)(27,28,32,31)(36,44)(40,47)(42,48)(45,46);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(17,19,22,24)(18,23,26,20)(21,27,30,32)(25,31,29,28)(33,34)(35,39)(36,40)(37,41)(38,43)(42,46)(44,47)(45,48);(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,31)(30,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,4,7,15)(2,9,11,5)(3,13,16,6)(8,10,12,14)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,29)(30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,7,9)(2,4,11,15)(3,14,16,10)(6,8,13,12)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27)(33,34)(35,38)(36,40)(37,41)(39,43)(42,45)(44,47)(46,48);(1,6,7,13)(2,10,11,14)(3,15,16,4)(5,8,9,12)(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8044389);