#290123

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 197319);