#8932822

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

3993 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8840018);