#477091

order	512
nilpotency class	3
isoclinism	isoclinic class of #476175
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384287);