#266746

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

763 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,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, 173942);