#8926722

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	#58989	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4123 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

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, 8833918);