#454170

order	512
nilpotency class	3
isoclinism	isoclinic class of #453974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361366);