#8504800

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

3945 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)(3,9)(4,8)(5,14)(6,11)(7,12)(10,15)(13,16)(19,20)(23,24)(27,28)(30,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3,7,4)(2,8,12,9)(5,11,15,13)(6,14,16,10)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3,7,4)(2,9,12,8)(5,13,15,11)(6,14,16,10)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,4,7,3)(2,9,12,8)(5,11,15,13)(6,14,16,10)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,5)(2,10)(3,13)(4,11)(6,9)(7,15)(8,16)(12,14)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30);(1,6)(2,11)(3,14)(4,10)(5,9)(7,16)(8,15)(12,13)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411996);