#8533652

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

3743 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,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, 8440848);