#128040

order	512
nilpotency class	4
isoclinism	isoclinic class of #7310
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1304
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	C₂×#1304
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240488, #72240492, #72240495, …
maximal normal subgroup of	#72240488, #72240492, #72240495, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35236);