#8492795

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

4083 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8399991);