#10023011

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

4168 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)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34)(35,41)(36,40)(37,42)(38,39)(43,48)(44,45)(46,47);(2,6)(3,9)(8,11)(10,12)(20,27)(23,25)(28,32)(30,31)(33,35)(34,37)(36,44)(38,42)(39,41)(40,48)(43,46)(45,47);(2,6)(3,9)(8,11)(10,12)(18,19)(23,25)(24,26)(30,31)(34,39)(35,42)(40,47)(43,44);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,46)(47,48);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36,38,46)(34,40,39,47)(35,43,42,44)(37,45,41,48);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,37)(34,35)(36,45)(38,41)(39,42)(40,43)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=gh,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9930207);