#8258580

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	#35133	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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)(3,8)(4,10)(5,12)(6,11)(7,9)(13,16)(14,15)(18,24)(19,20)(21,31)(27,30)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(1,3)(2,8)(4,6)(5,9)(7,12)(10,11)(13,14)(15,16)(17,18,23,24)(19,25,20,28)(21,32,31,22)(26,27,29,30)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(2,9)(3,12)(6,13)(10,15)(17,19,23,20)(18,25,24,28)(21,26,31,29)(22,30,32,27)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(1,4)(2,10)(3,13)(5,11)(6,12)(7,14)(8,16)(9,15)(17,20,23,19)(18,25,24,28)(21,29,31,26)(22,30,32,27)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(1,5)(2,3)(4,11)(6,15)(7,8)(9,12)(10,13)(14,16)(17,21)(18,22)(19,26)(20,29)(23,31)(24,32)(25,30)(27,28)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(1,6,7,13)(2,11,9,16)(3,4,12,14)(5,15,8,10)(17,22)(18,21)(19,27)(20,30)(23,32)(24,31)(25,29)(26,28)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=g,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8165776);