#10102214

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
prime radical	#55160	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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:

(1,2,6,8)(3,10,5,7)(4,9,14,13)(11,16,12,15)(17,18)(19,25)(20,24)(21,30)(22,26)(23,29)(27,32)(28,31)(33,34)(35,42)(36,41)(37,40)(38,39)(43,48)(44,47)(45,46);(1,3,6,5)(2,7,8,10)(4,12,14,11)(9,16,13,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(35,37)(40,42)(43,44)(47,48);(2,8)(7,10)(9,13)(15,16)(17,20,22,21)(18,24,26,30)(19,27,29,28)(23,31,25,32)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,2)(3,7)(4,13)(5,10)(6,8)(9,14)(11,16)(12,15)(17,21,22,20)(18,24,26,30)(19,28,29,27)(23,31,25,32)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,4,6,14)(2,9,8,13)(3,11,5,12)(7,15,10,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,29)(27,31)(28,32)(33,37,39,35)(34,40,38,42)(36,43,46,44)(41,48,45,47);(1,5,6,3)(2,10,8,7)(4,11,14,12)(9,15,13,16)(17,19)(18,25)(20,27)(21,28)(22,29)(23,26)(24,32)(30,31)(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=hi,c^2=gh,d^2=gh,e^2=h,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, 10009410);