#8746008

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3813 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)(3,11,13,8)(5,14,12,9)(6,10,16,15)(19,27)(20,29)(23,30)(24,32)(33,34)(35,42)(36,37)(38,40)(39,43)(41,48)(44,45)(46,47);(1,3,7,13)(2,8,4,11)(5,6,12,16)(9,10,14,15)(17,18,22,26)(19,23,27,30)(20,24,29,32)(21,25,28,31)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(2,4)(8,11)(9,14)(10,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,32)(26,30)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,4,7,2)(3,8,13,11)(5,14,12,9)(6,10,16,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,5)(2,9)(3,6)(4,14)(7,12)(8,10)(11,15)(13,16)(17,20,22,29)(18,24,26,32)(19,28,27,21)(23,31,30,25)(33,35,39,38)(34,42,43,40)(36,44,41,46)(37,45,48,47);(1,6)(2,10)(3,12)(4,15)(5,13)(7,16)(8,14)(9,11)(17,21,22,28)(18,25,26,31)(19,29,27,20)(23,32,30,24)(33,38,39,35)(34,40,43,42)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=i,d^2=gi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8653204);