#246322

order	512
nilpotency class	3
isoclinism	isoclinic class of #246319
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60371	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1915 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255902, #72256155, #72256161
maximal normal subgroup of	#72255902, #72256155, #72256161

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153518);