#10195500

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	#44632	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=gh,e^2=g,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, 10102696);