#478825

order	512
nilpotency class	3
isoclinism	isoclinic class of #476297
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

685 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:

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

Polycyclic presentation:

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

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, 386021);