#10196176

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

4039 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)(3,4)(5,13)(6,7)(8,10)(9,15)(11,14)(12,16)(17,18,22,23)(19,20,24,28)(21,29,31,25)(26,30,32,27)(33,34,38,41)(35,39,44,47)(36,37,45,40)(42,43,48,46);(2,7)(4,8)(9,13)(12,14)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,30)(31,32)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(2,7)(3,10)(9,13)(11,16)(18,23)(19,24)(25,29)(26,32)(33,36,38,45)(34,40,41,37)(35,42,44,48)(39,46,47,43);(1,3,6,10)(2,8,7,4)(5,11,15,16)(9,14,13,12)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,30)(31,32)(33,36)(34,37)(35,42)(38,45)(39,43)(40,41)(44,48)(46,47);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,20)(18,24)(19,23)(21,30)(22,28)(25,26)(27,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,21,22,31)(18,25,23,29)(19,26,24,32)(20,27,28,30)(33,37)(34,36)(35,43)(38,40)(39,42)(41,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=g,e^2=g,f^2=gi,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, 10103372);