#8455664

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	#40112	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(18,23)(19,28)(26,30)(27,32)(34,39)(35,42)(36,44)(45,47);(1,3,7,14)(2,8,11,12)(4,13,5,6)(9,16,15,10)(17,18)(19,24)(20,26)(21,27)(22,23)(25,28)(29,30)(31,32)(33,34,38,39)(35,37,42,40)(36,41,44,46)(43,45,48,47);(1,4,7,5)(2,9,11,15)(3,13,14,6)(8,16,12,10)(17,19)(18,24)(20,27)(21,26)(22,28)(23,25)(29,32)(30,31)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,5,7,4)(2,9,11,15)(3,6,14,13)(8,16,12,10)(17,19)(18,25)(20,27)(21,30)(22,28)(23,24)(26,31)(29,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,6)(2,10)(3,4)(5,14)(7,13)(8,9)(11,16)(12,15)(17,20,22,29)(18,26,23,30)(19,27,28,32)(21,25,31,24)(33,36)(34,41)(35,43)(37,45)(38,44)(39,46)(40,47)(42,48);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,21,22,31)(18,27,23,32)(19,26,28,30)(20,25,29,24)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8362860);