#8095270

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002466);