#8662825

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	#43283	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4155 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=g,e^2=g,f^2=gh,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, 8570021);