#453901

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

679 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361097);