#10410047

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	#57151	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4221 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:

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

Polycyclic presentation:

2^9|b^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10317243);