#8425624

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

3891 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=g,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^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, 8332820);