#457505

order	512
nilpotency class	3
isoclinism	isoclinic class of #456417
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

822 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:

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

Polycyclic presentation:

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

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, 364701);