#8755773

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=h,e^2=gi,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, 8662969);