#10171646

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

4060 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)(3,10)(4,7)(5,9)(6,8)(11,16)(12,14)(13,15)(17,18,22,25)(19,23,28,31)(20,21,29,24)(26,27,32,30)(33,34,38,41)(35,42,37,39)(36,40,45,48)(43,47,44,46);(1,3,7,6)(2,8,4,10)(5,13,14,11)(9,16,12,15)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(35,37)(39,42)(43,44)(46,47);(2,4)(8,10)(9,12)(15,16)(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,35,38,37)(34,39,41,42)(36,44,45,43)(40,47,48,46);(1,4,7,2)(3,10,6,8)(5,9,14,12)(11,15,13,16)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,36,38,45)(34,40,41,48)(35,43,37,44)(39,46,42,47);(1,5,7,14)(2,9,4,12)(3,11,6,13)(8,15,10,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37,38,35)(34,39,41,42)(36,43,45,44)(40,47,48,46);(1,6,7,3)(2,10,4,8)(5,11,14,13)(9,15,12,16)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48)

Polycyclic presentation:

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

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, 10078842);