#149037

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56233);