#418156

order	512
nilpotency class	3
isoclinism	isoclinic class of #417487
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	#35329	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1663 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318058, #72318275, #72318305
maximal normal subgroup of	#72318058, #72318275, #72318305

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325352);