#148751

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55947);