#10455717

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

4183 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10362913);