#8097041

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

4182 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)(5,9)(6,7)(8,10)(11,12)(13,16)(14,15)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30)(33,34,38,42)(35,39,37,41)(36,40,44,47)(43,46,45,48);(2,7)(3,10)(9,15)(11,16)(20,28)(24,31)(27,29)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,13)(12,15)(14,16)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32)(33,35,38,37)(34,39,42,41)(36,43,44,45)(40,46,47,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,13)(12,15)(14,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4,6,8)(2,3,7,10)(5,13,14,12)(9,16,15,11)(17,21,22,19)(18,23,25,26)(20,29,28,27)(24,30,31,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,48)(42,47);(1,5,6,14)(2,9,7,15)(3,11,10,16)(4,12,8,13)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,37,38,35)(34,41,42,39)(36,43,44,45)(40,46,47,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,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, 8004237);