#8186976

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

3990 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8094172);