#146444

order	512
nilpotency class	4
isoclinism	isoclinic class of #11677
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

946 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)(5,6,9,14,11,15,10,26)(7,16,18,33,21,29,20,35)(8,22,23,31,25,36,24,13)(12,27,28,19,30,34,32,17);(5,7,8,19,9,18,23,34,11,21,25,17,10,20,24,27)(6,12,13,29,26,32,36,33,15,30,31,16,14,28,22,35);(2,4)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239826, #72239831, #72239839, …
maximal normal subgroup of	#72239826, #72239831, #72239839, …
maximal quotient of	#72308291

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53640);