#8096280

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

3943 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,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, 8003476);