#8463189

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3902 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)(8,11)(10,15)(18,21)(19,20)(26,29)(27,28)(33,34,38,39)(35,41,44,40)(36,45,46,37)(42,43,48,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(34,39)(36,46)(40,41)(42,48);(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35,38,44)(34,40,39,41)(36,42,46,48)(37,43,45,47);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,36,38,46)(34,37,39,45)(35,42,44,48)(40,43,41,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,37)(34,36)(35,43)(38,45)(39,46)(40,42)(41,48)(44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=h,e^2=ghi,f^2=i,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, 8370385);