#8755231

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3773 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=ghi,e^2=h,f^2=g,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, 8662427);