#10410032

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	#49957	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,e^2=g,f^2=h,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, 10317228);