#8997132

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8904328);