#8630848

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	#54810	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538044);