#415982

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1576 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=i,d^2=hi,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, 323178);