#8127295

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4120 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=h,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, 8034491);