#454198

order	512
nilpotency class	3
isoclinism	isoclinic class of #453972
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
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

1537 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)(11,16)(12,26)(13,19)(14,18)(15,20)(17,27)(21,31)(22,39)(23,33)(24,32)(25,34)(28,35)(29,36)(30,37)(38,40);(1,2,4,5)(3,6,7,8)(9,11,15,25)(10,16,20,34)(12,22,28,37)(13,23,14,24)(17,31,36,40)(18,32,19,33)(21,29,38,27)(26,39,35,30);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,21)(13,29)(14,27)(15,28)(16,30)(18,35)(19,26)(20,36)(22,23)(24,37)(25,38)(31,33)(32,40)(34,39);(3,7)(6,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,31)(18,26)(19,35)(20,36)(21,23)(24,38)(25,37)(30,33)(32,39)(34,40)

Polycyclic presentation:

2^9|b^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,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, 361394);