#8994401

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	#59674	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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)(3,8)(4,10)(5,11)(6,14)(7,9)(12,16)(13,15)(17,18,22,26)(19,20,24,23)(21,25,29,32)(27,28,31,30);(1,3,7,4)(2,8,9,10)(5,14,15,12)(6,13,16,11)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34,38,42)(35,40,45,39)(36,37,41,48)(43,47,46,44);(2,9)(3,4)(5,15)(6,16)(19,24)(20,23)(21,29)(25,32)(33,35,38,45)(34,39,42,40)(36,46,41,43)(37,47,48,44);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,19)(18,23)(20,26)(21,27)(22,24)(25,30)(28,32)(29,31)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,44,40,47);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,20,22,23)(18,24,26,19)(21,28,29,30)(25,31,32,27)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,21,22,29)(18,25,26,32)(19,27,24,31)(20,28,23,30)(33,37)(34,41)(35,44)(36,42)(38,48)(39,46)(40,43)(45,47)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=i,e^2=gh,f^2=g,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, 8901597);