#8756081

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	#47697	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8663277);