#8188570

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

3998 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=ghi,e^2=h,f^2=g,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, 8095766);