#8471028

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

3967 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=hi,e^2=ghi,f^2=gi,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, 8378224);