#8752205

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	#56210	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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)(7,10)(11,15)(14,16)(17,18)(19,28)(20,31)(21,22)(23,24)(25,29)(26,30)(27,32)(35,42)(38,40)(43,48)(45,47);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(18,24)(19,29)(21,27)(26,31)(33,34,39,36)(35,40,42,38)(37,44,46,41)(43,45,48,47);(2,6)(7,10)(8,12)(14,16)(17,19)(18,25)(20,27)(21,30)(22,31)(23,29)(24,28)(26,32)(33,35,39,42)(34,40,36,38)(37,43,46,48)(41,47,44,45);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,30)(18,26,24,31)(19,27,29,21)(22,25,32,28)(33,36,39,34)(35,38,42,40)(37,44,46,41)(43,45,48,47);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21,23,27)(18,22,24,32)(19,30,29,20)(25,31,28,26)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,27)(19,31)(20,25)(21,24)(23,32)(26,29)(28,30)(33,38)(34,35)(36,42)(37,45)(39,40)(41,43)(44,48)(46,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,e^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, 8659401);