#8630898

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	#55335	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4013 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538094);