#457319

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

904 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)(9,10,13,18,15,19,14,29)(11,16,21,31,22,32,23,39)(12,24,26,35,27,34,28,17)(20,36,37,33,25,40,38,30);(2,5)(6,8)(9,11,13,21,15,22,14,23)(10,16,18,31,19,32,29,39)(12,25,26,38,27,20,28,37)(17,33,24,40,35,30,34,36);(2,5)(3,7)(9,12)(10,17)(11,20)(13,28)(14,26)(15,27)(16,30)(18,34)(19,35)(21,38)(22,25)(23,37)(24,29)(31,40)(32,33)(36,39);(1,3,4,7)(2,6,5,8)(9,10,15,19)(11,16,22,32)(12,17,27,35)(13,18,14,29)(20,30,25,33)(21,31,23,39)(24,28,34,26)(36,38,40,37)

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,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, 364515);