#8505270

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3884 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=ghi,e^2=h,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, 8412466);