#320737

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 227933);