#459304

order	512
nilpotency class	3
isoclinism	isoclinic class of #454369
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23406	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1153 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)(3,7)(4,5)(6,8)(9,10,15,20)(11,21,26,37)(12,27,29,17)(13,30,31,18)(14,32,22,19)(16,33,36,25)(23,38,28,35)(24,40,39,34);(2,5)(6,7)(9,11,22,33,15,26,14,25)(10,16,32,37,20,36,19,21)(12,28,31,39,29,23,13,24)(17,34,18,35,27,40,30,38);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,23)(13,22)(14,31)(15,29)(16,34)(18,19)(20,27)(21,35)(24,25)(26,28)(30,32)(33,39)(36,40)(37,38);(3,8)(6,7)(9,13,22,12,15,31,14,29)(10,18,32,27,20,30,19,17)(11,24,33,28,26,39,25,23)(16,35,37,40,36,38,21,34)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^c=hi,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, 366500);