#146572

order	512
nilpotency class	4
isoclinism	isoclinic class of #11549
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂²	C₂×#342
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1724 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)(5,6,9,13,10,14,23,31)(7,15,17,32,18,27,21,35)(8,12,19,28,20,29,22,30)(11,24,25,33,26,34,36,16);(5,7)(6,11,14,26)(8,16,20,33)(9,21,23,17)(10,18)(12,27)(13,25)(15,29)(19,34)(22,24)(28,35,30,32)(31,36);(2,4)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,d^b=dh,f^b=fhi,h^b=hi,f^d=fi,g^d=gi,g^f=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280625, #72280657, #72280792, …
maximal normal subgroup of	#72280625, #72280657, #72280792, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53768);