#8679894

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8587090);