#8810083

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	#44223	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717279);