#8461994

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

4103 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

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, 8369190);