#8746007

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

3807 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)(3,4)(5,15)(6,11)(7,9)(8,13)(10,12)(14,16)(19,27)(20,29)(23,30)(24,32)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(1,3,7,13)(2,8,9,4)(5,6,12,16)(10,11,15,14)(17,18,22,26)(19,23,27,30)(20,24,29,32)(21,25,28,31)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(2,9)(4,8)(10,15)(11,14)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,32)(26,30)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,4,7,8)(2,3,9,13)(5,14,12,11)(6,15,16,10)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,10)(3,6)(4,14)(7,12)(8,11)(9,15)(13,16)(17,20,22,29)(18,24,26,32)(19,28,27,21)(23,31,30,25)(33,35,39,38)(34,42,36,40)(37,43,46,45)(41,48,44,47);(1,6)(2,11)(3,12)(4,10)(5,13)(7,16)(8,15)(9,14)(17,21,22,28)(18,25,26,31)(19,29,27,20)(23,32,30,24)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)

Polycyclic presentation:

2^9|a^2=i,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, 8653203);