#8096709

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(2,8)(4,9)(10,15)(12,16)(17,18,24,28)(19,23,25,20)(21,29,32,26)(22,27,30,31);(1,2)(3,4)(5,9)(6,10)(7,8)(11,12)(13,16)(14,15)(17,19,24,25)(18,20,28,23)(21,30,32,22)(26,31,29,27)(34,38)(39,45)(40,46)(47,48);(1,3,7,5)(2,4,8,9)(6,11,14,13)(10,12,15,16)(17,20,24,23)(18,25,28,19)(21,27,32,31)(22,26,30,29)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46);(1,4,7,9)(2,3,8,5)(6,12,14,16)(10,11,15,13)(17,21)(18,26)(19,22)(20,27)(23,31)(24,32)(25,30)(28,29);(1,5,7,3)(2,9,8,4)(6,11,14,13)(10,12,15,16)(17,22)(18,27)(19,21)(20,26)(23,29)(24,30)(25,32)(28,31)(33,35,37,42)(34,39,38,45)(36,43,44,41)(40,48,46,47);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,23,24,20)(18,25,28,19)(21,31,32,27)(22,26,30,29)(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=gi,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, 8003905);