#8718381

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3871 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)(3,11,12,8)(4,13,14,9)(5,10,15,16)(17,18,23,27)(19,24,28,31)(20,25,29,30)(21,22,26,32)(34,36)(39,42)(40,44)(46,48);(2,7)(3,12)(9,13)(10,16)(17,19,23,28)(18,24,27,31)(20,21,29,26)(22,30,32,25)(35,41)(39,42)(43,47)(46,48);(2,7)(3,12)(9,13)(10,16)(19,28)(20,29)(24,31)(25,30)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,9,15,13)(17,20,23,29)(18,25,27,30)(19,21,28,26)(22,31,32,24)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,8,15,11)(17,21)(18,22)(19,29)(20,28)(23,26)(24,30)(25,31)(27,32)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,5,6,15)(2,10,7,16)(3,13,12,9)(4,11,14,8)(17,22)(18,26)(19,30)(20,24)(21,27)(23,32)(25,28)(29,31)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

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

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, 8625577);