#8806883

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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)(3,10)(9,13)(11,16)(19,22)(24,26)(27,29)(31,32)(33,34,39,37)(35,42,41,38)(36,44,46,40)(43,45,47,48);(1,2,6,7)(3,8,10,4)(5,13,14,9)(11,12,16,15)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31)(34,37)(35,41)(40,44)(43,47);(3,10)(4,8)(11,16)(12,15)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35)(34,38)(36,43)(37,42)(39,41)(40,45)(44,48)(46,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,13)(14,16)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,37,39,34)(35,42,41,38)(36,44,46,40)(43,48,47,45);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,38,39,42)(34,41,37,35)(36,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=hi,e^2=ghi,f^2=h,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, 8714079);