#128224

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1745
prime radical	C₂³	C₄×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=d,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=cg,d^a=di,f^a=fi,c^b=chi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224829, #72224925, #72224927
maximal normal subgroup of	#72224829, #72224925, #72224927

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35420);