#8445419

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

4192 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=h,e^2=hi,f^2=h,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, 8352615);