#8096958

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

4256 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:

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

Polycyclic presentation:

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

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, 8004154);