#8371262

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3833 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:

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

Polycyclic presentation:

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

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, 8278458);