#8202431

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	#37891	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8109627);