#9851021

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

3943 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9758217);