#437766

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1274 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:

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

Polycyclic presentation:

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

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, 344962);