#246326

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	#32890	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1910 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)(12,17)(13,25)(14,27)(15,19)(16,21)(18,24)(22,30)(23,32)(26,33)(28,31)(29,34)(35,37)(36,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,23)(10,16,19,31)(12,21,14,28)(13,22,26,38)(17,32,27,20)(18,30,34,40)(24,36,29,37)(25,35,33,39);(1,3)(2,5)(4,7)(6,8)(9,12,15,14)(10,17,19,27)(11,21,23,28)(13,24,26,29)(16,20,31,32)(18,33,34,25)(22,36,38,37)(30,35,40,39);(1,2)(3,5)(4,6)(7,8)(9,13,15,26)(10,18,19,34)(11,22,23,38)(12,24,14,29)(16,30,31,40)(17,33,27,25)(20,35,32,39)(21,36,28,37)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^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, #72256151, #72256161
maximal normal subgroup of	#72255902, #72256151, #72256161

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153522);