#8127329

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	#41474	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034525);