#10102232

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

4041 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=gh,e^2=gh,f^2=h,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, 10009428);