#317176

order	512
nilpotency class	3
isoclinism	isoclinic class of #317171
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	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224372);