#8093416

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

4294 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,16)(13,15)(20,28)(24,29)(26,32)(30,31)(33,34,38,36)(35,41,42,39)(37,44,45,40)(43,46,48,47);(2,7)(4,9)(8,10)(12,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(34,36)(35,42)(40,44)(43,48);(3,11)(8,10)(12,15)(13,16)(18,22)(23,25)(24,29)(30,31)(33,35,38,42)(34,39,36,41)(37,43,45,48)(40,46,44,47);(1,3)(2,8)(4,12)(5,13)(6,11)(7,10)(9,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35)(34,39)(36,41)(37,43)(38,42)(40,46)(44,47)(45,48);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,38,34)(35,41,42,39)(37,44,45,40)(43,47,48,46);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000612);