#8533631

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

3970 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:

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

Polycyclic presentation:

2^9|a^2=hi,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, 8440827);