#8095771

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

4073 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)(3,10)(4,13)(5,14)(6,8)(7,12)(9,15)(11,16)(21,29)(26,30)(27,32)(28,31)(33,34)(35,40)(36,42)(37,41)(38,43)(39,46)(44,47)(45,48);(1,3)(2,7)(4,14)(5,9)(6,12)(8,10)(11,15)(13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,47)(42,48);(2,8)(3,12)(9,13)(11,14)(18,23)(20,24)(26,30)(28,31)(33,35)(34,40)(36,44)(37,45)(38,46)(39,43)(41,48)(42,47);(1,4)(2,9)(3,11)(5,10)(6,15)(7,16)(8,13)(12,14)(17,19,22,25)(18,24,23,20)(21,27,29,32)(26,31,30,28)(33,36,38,37)(34,41,43,42)(35,44,46,45)(39,47,40,48);(1,3)(2,10)(4,11)(5,9)(6,12)(7,8)(13,16)(14,15)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,34)(35,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5)(2,11)(3,9)(4,10)(6,16)(7,15)(8,14)(12,13)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,37,38,36)(34,42,43,41)(35,45,46,44)(39,48,40,47)

Polycyclic presentation:

2^9|d^2=gi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002967);