#8662827

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	#48218	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4144 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)(3,12)(4,10)(5,7)(6,9)(8,11)(13,15)(14,16)(17,18)(19,27)(20,26)(21,25)(22,23)(24,29)(28,31)(30,32)(33,34,38,40)(35,36,42,44)(37,39,47,46)(41,43,45,48);(1,3,8,6)(2,9,11,12)(4,13,5,14)(7,15,10,16)(17,19)(18,24)(20,28)(21,30)(22,27)(23,29)(25,31)(26,32)(33,35,38,42)(34,36,40,44)(37,41,47,45)(39,43,46,48);(1,4,8,5)(2,10,11,7)(3,13,6,14)(9,16,12,15)(17,20,23,21)(18,25,22,26)(19,28,29,30)(24,31,27,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,46)(47,48);(1,5,8,4)(2,10,11,7)(3,14,6,13)(9,16,12,15)(17,21,23,20)(18,26,22,25)(19,28,29,30)(24,31,27,32)(33,35)(34,36)(37,45)(38,42)(39,48)(40,44)(41,47)(43,46);(1,6,8,3)(2,9,11,12)(4,14,5,13)(7,15,10,16)(17,22,23,18)(19,27,29,24)(20,26,21,25)(28,32,30,31)(33,34)(35,36)(37,46)(38,40)(39,47)(41,48)(42,44)(43,45);(1,7,8,10)(2,5,11,4)(3,15,6,16)(9,14,12,13)(17,19)(18,27)(20,28)(21,30)(22,24)(23,29)(25,32)(26,31)(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=gh,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, 8570023);