#8097026

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

4198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004222);