#10147070

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
prime radical	#44236	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4213 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)(3,11)(4,13)(5,10)(6,14)(7,9)(8,12)(15,16)(17,18,23,25)(19,20,28,24)(21,26,32,30)(22,29,31,27)(33,34,38,41)(35,37,43,39)(36,40,44,47)(42,45,48,46);(1,3,7,12)(2,8,9,11)(4,5,14,15)(6,10,13,16)(17,19,23,28)(18,24,25,20)(21,31,32,22)(26,29,30,27)(36,44)(40,47)(42,48)(45,46);(2,9)(3,12)(4,14)(10,16)(18,25)(20,24)(26,30)(27,29)(33,35,38,43)(34,39,41,37)(36,42,44,48)(40,46,47,45);(1,4)(2,6)(3,5)(7,14)(8,10)(9,13)(11,16)(12,15)(17,20)(18,19)(21,27)(22,30)(23,24)(25,28)(26,31)(29,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47);(1,5)(2,10)(3,4)(6,8)(7,15)(9,16)(11,13)(12,14)(17,21,23,32)(18,26,25,30)(19,22,28,31)(20,29,24,27)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,6)(2,4)(3,10)(5,8)(7,13)(9,14)(11,15)(12,16)(17,22)(18,27)(19,21)(20,30)(23,31)(24,26)(25,29)(28,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,e^2=gi,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, 10054266);