#8810088

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

3914 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)(11,16)(17,18,23,21)(19,24,29,31)(20,26,30,25)(22,32,28,27);(1,2,6,7)(3,4,9,12)(5,14,13,8)(10,16,15,11)(17,19,23,29)(18,24,21,31)(20,22,30,28)(25,27,26,32)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(1,3,6,9)(2,4,7,12)(5,10,13,15)(8,11,14,16)(17,20,23,30)(18,25,21,26)(19,28,29,22)(24,32,31,27)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,3)(2,4)(5,10)(6,9)(7,12)(8,11)(13,15)(14,16)(17,20)(18,26)(19,22)(21,25)(23,30)(24,32)(27,31)(28,29)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,16)(14,15)(17,21,23,18)(19,24,29,31)(20,25,30,26)(22,32,28,27)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,5,6,13)(2,8,7,14)(3,10,9,15)(4,11,12,16)(17,22)(18,27)(19,20)(21,32)(23,28)(24,25)(26,31)(29,30)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8717284);