#146358

order	512
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂×C₄	C₄³
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	C₄×#99
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=i,g^2=i,b^a=bd,d^a=dh,f^a=fi,h^a=hi,e^b=ehi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240802, #72257236, #72257245
maximal normal subgroup of	#72240802, #72257236, #72257245
maximal quotient of	#72240806, #72317540

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53554);