#411023

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

779 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318219);