#405949

order	512
nilpotency class	3
isoclinism	isoclinic class of #405949
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14921
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1332 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)(3,11,14,7)(4,8,17,23)(5,18,19,9)(6,10)(12,20)(13,21)(15,24)(16,22)(25,27)(26,31,28,32)(29,30);(1,3)(2,7)(4,15,17,12)(5,13)(6,14)(8,20,23,24)(9,21)(10,11)(16,28,29,26)(18,25)(19,27)(22,31,30,32);(1,4)(2,8,10,23)(3,12)(5,16)(6,17)(7,20,11,24)(9,22,18,30)(13,26)(14,15)(19,29)(21,31,25,32)(27,28);(1,5)(2,9)(3,13)(4,16)(6,19)(7,21)(8,22)(10,18)(11,25)(12,26)(14,27)(15,28)(17,29)(20,31)(23,30)(24,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237784, #72237785, #72237789, …
maximal normal subgroup of	#72237784, #72237785, #72237789, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313145);