#395560

order	512
nilpotency class	3
isoclinism	isoclinic class of #395560
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14725
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3021	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2137 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241162, #72241168, #72241303, …
maximal normal subgroup of	#72241162, #72241168, #72241303, …
maximal quotient of	C₂×#395560

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302756);