#8097072

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

4159 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004268);