#8533624

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440820);