#8931369

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	#55754	C₂
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:

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

Polycyclic presentation:

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

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, 8838565);