#8810327

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	#45806	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4136 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,d^2=g,f^2=gi,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, 8717523);