#8718072

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4010 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)(3,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,18,23,27)(19,24,28,31)(20,25,29,30)(21,22,26,32)(33,34)(35,42)(36,39)(37,44)(38,40)(41,46)(43,47)(45,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,23,28)(18,24,27,31)(20,21,29,26)(22,30,32,25)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(4,13)(7,12)(10,16)(14,15)(19,28)(20,29)(24,31)(25,30)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,4)(2,7)(3,10)(5,13)(6,14)(8,15)(9,12)(11,16)(17,20,23,29)(18,25,27,30)(19,21,28,26)(22,31,32,24)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,21)(18,22)(19,29)(20,28)(23,26)(24,30)(25,31)(27,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,2)(3,6)(4,12)(5,9)(7,13)(8,11)(10,15)(14,16)(17,22)(18,26)(19,30)(20,24)(21,27)(23,32)(25,28)(29,31)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8625268);