#8750822

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3803 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)(3,9)(4,7)(5,8)(6,11)(10,16)(12,14)(13,15)(19,21)(20,24)(27,29)(28,30)(33,34,39,42)(35,37,40,36)(38,46,47,41)(43,44,48,45);(1,2,5,8)(3,6,11,9)(4,7,14,12)(10,15,13,16)(17,18,23,26)(19,20,21,24)(22,25,31,32)(27,28,29,30)(33,35,39,40)(34,36,42,37)(38,43,47,48)(41,44,46,45);(1,2,5,8)(3,6,11,9)(4,7,14,12)(10,15,13,16)(17,19,23,21)(18,24,26,20)(22,27,31,29)(25,30,32,28)(33,36,39,37)(34,40,42,35)(38,44,47,45)(41,48,46,43);(1,2)(3,6)(4,12)(5,8)(7,14)(9,11)(10,16)(13,15)(17,20,23,24)(18,19,26,21)(22,30,31,28)(25,29,32,27)(33,37,39,36)(34,35,42,40)(38,45,47,44)(41,43,46,48);(1,3)(2,6)(4,13)(5,11)(7,16)(8,9)(10,14)(12,15)(17,21,23,19)(18,24,26,20)(22,27,31,29)(25,28,32,30)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46);(1,4,5,14)(2,7,8,12)(3,10,11,13)(6,15,9,16)(17,22,23,31)(18,25,26,32)(19,27,21,29)(20,28,24,30)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658018);