#8505587

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3892 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=i,e^2=h,f^2=hi,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, 8412783);