#8906971

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	#48933	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8814167);