#8751369

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

3797 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=ghi,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, 8658565);