#8127462

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,e^2=h,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, 8034658);