#8603553

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	#55705	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4040 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=gh,e^2=gh,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, 8510749);