#8755452

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	#46628	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3983 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662648);