#8108780

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	#53204	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015976);