#10202110

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	#49331	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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)(3,9)(4,5)(6,10)(7,13)(8,14)(11,15)(12,16)(17,18,23,28)(19,27,22,24)(20,30,31,25)(21,26,32,29)(33,34)(35,39)(36,40)(37,48)(38,42)(41,45)(43,44)(46,47);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19,23,22)(18,24,28,27)(20,29,31,26)(21,30,32,25)(33,35,38,44)(34,39,42,43)(36,45,47,37)(40,41,46,48);(17,20,23,31)(18,25,28,30)(19,29,22,26)(21,27,32,24)(33,34,38,42)(35,43,44,39)(36,40,47,46)(37,41,45,48);(1,4,6,14)(2,8,10,5)(3,11,13,16)(7,15,9,12)(17,20)(18,25)(19,29)(21,24)(22,26)(23,31)(27,32)(28,30)(33,34)(35,39)(36,46)(37,41)(38,42)(40,47)(43,44)(45,48);(1,5,6,8)(2,4,10,14)(3,12,13,15)(7,11,9,16)(17,21)(18,26)(19,25)(20,27)(22,30)(23,32)(24,31)(28,29)(33,36,38,47)(34,40,42,46)(35,37,44,45)(39,48,43,41);(1,3)(2,9)(4,11)(5,15)(6,13)(7,10)(8,12)(14,16)(17,22,23,19)(18,27,28,24)(20,26,31,29)(21,25,32,30)(33,37)(34,41)(35,36)(38,45)(39,46)(40,43)(42,48)(44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=ghi,e^2=gh,f^2=hi,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, 10109306);