#8295653

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

3843 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,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=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, 8202849);