#316996

order	512
nilpotency class	3
isoclinism	isoclinic class of #316991
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224192);