#468432

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

643 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=g,c^2=g,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^b=hi,d^c=de,f^c=fi,e^d=ei,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, 375628);