#9040848

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

3994 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,16)(14,15)(33,34,39,43)(35,40,38,42)(36,46,37,41)(44,48,45,47);(2,7)(3,12)(4,10)(14,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,35,39,38)(34,40,43,42)(36,45,37,44)(41,48,46,47);(4,10)(5,9)(13,15)(14,16)(18,22)(20,29)(23,25)(26,30)(33,36,39,37)(34,41,43,46)(35,44,38,45)(40,47,42,48);(1,3)(2,8)(4,15)(5,14)(6,12)(7,11)(9,16)(10,13)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,37,39,36)(34,41,43,46)(35,45,38,44)(40,47,42,48);(1,4,6,10)(2,9,7,5)(3,13,12,15)(8,14,11,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,38,39,35)(34,42,43,40)(36,44,37,45)(41,47,46,48);(1,5,6,9)(2,10,7,4)(3,14,12,16)(8,13,11,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,34)(35,40)(36,46)(37,41)(38,42)(39,43)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=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, 8948044);