#8127231

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4156 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=g,e^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, 8034427);