#300409

order	512
nilpotency class	3
isoclinism	isoclinic class of #300397
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3559	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

849 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 207605);