#8508897

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=gi,e^2=ghi,f^2=gi,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, 8416093);