#8754991

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3711 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)(4,11)(5,9)(6,7)(10,15)(12,16)(13,14)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,34,40,38)(35,43,47,41)(36,37,42,39)(44,45,48,46);(2,7)(4,8)(5,13)(10,16)(17,19,22,18)(20,25,28,23)(21,24,29,26)(27,30,32,31)(33,35,40,47)(34,41,38,43)(36,48,42,44)(37,45,39,46);(1,3,6,11)(2,8,7,4)(5,10,13,16)(9,15,14,12)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,36)(34,39)(35,44)(37,38)(40,42)(41,45)(43,46)(47,48);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,16)(14,15)(33,37,40,39)(34,36,38,42)(35,45,47,46)(41,48,43,44);(1,4)(2,3)(5,12)(6,8)(7,11)(9,10)(13,15)(14,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38,40,34)(35,41,47,43)(36,37,42,39)(44,45,48,46);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(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=h,c^2=gi,d^2=i,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, 8662187);