#6382543

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60557	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1787 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ci,d^b=df,g^c=gi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313507, #72315597, #72315625
maximal normal subgroup of	#72313507, #72315597, #72315625

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289739);