#436315

order	512
nilpotency class	3
isoclinism	isoclinic class of #421903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

788 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343511);