#10420055

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

4138 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=gh,e^2=g,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, 10327251);