#8505741

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3881 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=g,e^2=i,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, 8412937);