#356007

order	512
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1570 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)(4,6)(5,8)(9,10,13,18,14,17,25,31)(11,15,21,28,20,27,35,39)(12,16,23,29,22,30,24,32)(19,26,34,38,33,37,36,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,26)(17,27)(18,28)(22,33)(23,34)(24,36)(25,35)(29,38)(30,37)(31,39)(32,40);(3,8)(5,7)(9,12)(10,16,17,30)(11,19)(13,24)(14,22)(15,26,27,37)(18,32,31,29)(20,33)(21,36)(23,25)(28,40,39,38)(34,35);(2,6)(5,7)(11,20)(15,27)(19,33)(21,35)(26,37)(28,39)(34,36)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303331, #72303582, #72303583
maximal normal subgroup of	#72303331, #72303582, #72303583

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263203);