#382706

order	512
nilpotency class	3
isoclinism	isoclinic class of #188768
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1850 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22)(13,17)(14,18)(16,24)(19,33)(20,28)(21,29)(23,36)(25,37)(26,32)(27,34)(30,38)(31,39)(35,40);(2,5)(6,8)(9,11)(10,15)(12,23,24,38)(13,21)(14,20)(16,30,22,36)(17,29)(18,28)(19,25,34,39)(26,35)(27,31,33,37)(32,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16,18,22)(11,19,20,34)(13,25)(14,24)(15,27)(17,31,32,37)(21,23,35,38)(26,39)(28,33)(29,30)(36,40);(3,7)(6,8)(11,20)(15,28)(19,34)(21,35)(23,38)(27,33)(29,40)(30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318255, #72318259, #72318340
maximal normal subgroup of	#72318255, #72318259, #72318340

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289902);