#8096334

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

3882 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)(3,10)(4,12)(5,8)(6,9)(7,14)(11,13)(15,16)(17,18,24,26)(19,20,23,22)(21,29,32,25)(27,31,30,28);(1,3,6,8)(2,5,9,10)(4,11,14,15)(7,13,12,16)(17,19,24,23)(18,22,26,20)(21,30,32,27)(25,28,29,31)(34,38)(39,45)(40,46)(47,48);(17,20,24,22)(18,19,26,23)(21,28,32,31)(25,27,29,30)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46);(1,4)(2,7)(3,11)(5,13)(6,14)(8,15)(9,12)(10,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,30)(24,32)(26,29);(1,2)(3,5)(4,7)(6,9)(8,10)(11,13)(12,14)(15,16)(17,22,24,20)(18,19,26,23)(21,31,32,28)(25,27,29,30)(33,35,37,42)(34,39,38,45)(36,43,44,41)(40,48,46,47);(1,5)(2,8)(3,9)(4,13)(6,10)(7,15)(11,12)(14,16)(17,23,24,19)(18,22,26,20)(21,30,32,27)(25,31,29,28)(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=g,c^2=h,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, 8003530);