#8184915

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

4126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8092111);