#8838427

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3885 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8745623);