#8188026

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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)(18,20)(19,25)(24,28)(27,31)(34,38)(35,41)(40,43)(42,47);(4,12)(9,14)(11,13)(15,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(34,38)(39,45)(40,43)(46,48);(1,3,7,5)(2,8,6,10)(4,11,12,13)(9,15,14,16)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,34,37,38)(35,39,41,45)(36,43,44,40)(42,48,47,46);(1,4,7,12)(2,9,6,14)(3,11,5,13)(8,15,10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(1,5,7,3)(2,10,6,8)(4,13,12,11)(9,16,14,15)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,35,37,41)(34,39,38,45)(36,42,44,47)(40,46,43,48);(1,6,7,2)(3,10,5,8)(4,14,12,9)(11,16,13,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48)

Polycyclic presentation:

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

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, 8095222);