#405219

order	512
nilpotency class	3
isoclinism	isoclinic class of #394026
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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

2228 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)(13,14,19,24)(15,25,29,36)(16,30,34,39)(17,23,18,22)(20,28,37,27)(21,33,40,32)(26,41,43,35)(31,38,42,44);(2,4)(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,31)(17,28)(18,27)(19,29)(21,38)(22,36)(23,25)(24,37)(26,33)(30,41)(32,43)(34,42)(35,39)(40,44);(13,16,19,34)(14,21,24,40)(15,26,29,43)(17,33,18,32)(20,35,37,41)(22,39,23,30)(25,38,36,44)(27,31,28,42);(6,10)(9,12)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=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, 312415);