#10411856

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

4313 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)(3,9)(4,11)(5,13)(6,7)(8,15)(10,12)(14,16)(17,18,23,26)(19,24,20,21)(22,25,30,32)(27,31,28,29)(33,34)(35,40)(36,46)(37,47)(38,41)(39,45)(42,48)(43,44);(2,7)(4,9)(8,13)(12,16)(17,19,23,20)(18,21,26,24)(22,27,30,28)(25,29,32,31)(33,35)(34,39)(36,43)(37,44)(38,45)(40,41)(42,47)(46,48);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,19,23,20)(18,24,26,21)(22,27,30,28)(25,31,32,29)(33,36)(34,37)(35,42)(38,47)(39,48)(40,44)(41,46)(43,45);(1,3)(2,4)(5,14)(6,11)(7,9)(8,16)(10,15)(12,13)(17,20,23,19)(18,24,26,21)(22,27,30,28)(25,29,32,31)(33,36)(34,37)(35,43)(38,47)(39,44)(40,48)(41,46)(42,45);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,14)(15,16)(17,21,23,24)(18,20,26,19)(22,29,30,31)(25,28,32,27)(33,35)(34,40)(36,42)(37,44)(38,45)(39,41)(43,47)(46,48);(1,5)(2,8)(3,10)(4,12)(6,15)(7,13)(9,16)(11,14)(17,22,23,30)(18,25,26,32)(19,27,20,28)(21,29,24,31)(33,37)(34,36)(35,44)(38,46)(39,43)(40,42)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,e^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, 10319052);