#8096708

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

3886 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003904);