#415980

order	512
nilpotency class	3
isoclinism	isoclinic class of #394335
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 323176);