#8127501

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3924 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8034697);