#146975

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1954
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210510
maximal normal subgroup of	#72210510
maximal quotient of	#72210441

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54171);