#8417096

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

3769 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324292);