#8340075

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 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)(3,11)(4,14)(5,8)(6,12)(7,9)(10,15)(13,16)(17,18,23,20)(19,24,22,26)(21,25,30,28)(27,31,29,32)(33,34,39,38)(35,42,41,37)(36,40,44,46)(43,48,47,45);(1,3,7,5)(2,8,9,11)(4,12,15,13)(6,14,16,10)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(36,44)(40,46)(43,47)(45,48);(2,9)(6,16)(8,11)(10,14)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,47)(46,48);(1,4,7,15)(2,10,9,14)(3,13,5,12)(6,8,16,11)(17,20,23,18)(19,26,22,24)(21,25,30,28)(27,31,29,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,44)(41,47)(42,48);(1,5,7,3)(2,11,9,8)(4,13,15,12)(6,10,16,14)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,39,42)(34,41,38,35)(36,45,44,48)(40,47,46,43);(1,6,7,16)(2,12,9,13)(3,14,5,10)(4,8,15,11)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,38,39,34)(35,42,41,37)(36,46,44,40)(43,48,47,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=i,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8247271);