#8928022

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	#36814	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18)(19,27)(20,22)(21,25)(23,26)(24,29)(28,32)(30,31)(34,36)(35,40)(38,46)(45,48);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,19,23,29)(18,24,26,27)(20,28,30,25)(21,31,32,22)(33,34,39,36)(35,42,40,37)(38,41,46,44)(43,48,47,45);(3,11)(4,12)(6,16)(8,13)(17,20,23,30)(18,22,26,31)(19,28,29,25)(21,27,32,24)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,48)(46,47);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,20)(18,22)(19,28)(21,24)(23,30)(25,29)(26,31)(27,32)(33,36,39,34)(35,42,40,37)(38,44,46,41)(43,48,47,45);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,21)(18,25)(19,22)(20,27)(23,32)(24,30)(26,28)(29,31)(33,37,39,42)(34,40,36,35)(38,45,46,48)(41,47,44,43);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,22)(18,20)(19,21)(23,31)(24,28)(25,27)(26,30)(29,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

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

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, 8835218);