#8445745

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	#2828	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8352941);