#8107236

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

4119 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014432);