#8492790

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

4095 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,e^2=gi,f^2=g,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, 8399986);