#10102249

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

4081 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)(5,8)(6,7)(10,14)(12,13)(15,16)(17,18,24,26)(19,27,23,25)(20,22,21,31)(28,32,29,30)(33,34)(35,42)(36,41)(37,40)(38,39)(43,48)(44,47)(45,46);(1,3)(2,4)(5,13)(6,11)(7,9)(8,14)(10,15)(12,16)(17,19,24,23)(18,25,26,27)(20,28,21,29)(22,30,31,32)(35,37)(40,42)(43,44)(47,48);(2,7)(4,9)(8,16)(12,14)(17,20,24,21)(18,22,26,31)(19,28,23,29)(25,30,27,32)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,4)(2,3)(5,14)(6,9)(7,11)(8,13)(10,16)(12,15)(17,21,24,20)(18,22,26,31)(19,29,23,28)(25,30,27,32)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,5,6,15)(2,8,7,16)(3,10,11,13)(4,12,9,14)(17,22)(18,21)(19,30)(20,26)(23,32)(24,31)(25,29)(27,28)(33,37,39,35)(34,40,38,42)(36,43,46,44)(41,48,45,47);(1,3)(2,4)(5,13)(6,11)(7,9)(8,14)(10,15)(12,16)(17,23,24,19)(18,25,26,27)(20,29,21,28)(22,30,31,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10009445);