#8757090

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	#48173	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3926 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,e^2=gi,f^2=gh,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, 8664286);