#405368

order	512
nilpotency class	3
isoclinism	isoclinic class of #394335
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312564);