#474177

order	512
nilpotency class	3
isoclinism	isoclinic class of #457592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

705 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)(9,10,15,18,14,19,13,27)(11,20,29,31,21,37,23,16)(12,17,28,33,26,34,25,35)(22,38,40,32,24,36,39,30);(1,2,4,5)(3,6,7,8)(11,21)(22,24)(23,29)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11,13,23,14,21,15,29)(10,16,18,20,19,31,27,37)(12,24,25,40,26,22,28,39)(17,32,33,36,34,30,35,38);(3,7)(6,8)(9,12,14,26)(10,17,19,34)(11,22,21,24)(13,28,15,25)(16,30,31,32)(18,35,27,33)(20,36,37,38)(23,40,29,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=egh,c^2=f,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,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, 381373);