#449959

order	512
nilpotency class	3
isoclinism	isoclinic class of #449959
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15954
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

1173 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)(4,15,18,8)(5,9)(6,10)(7,14)(12,26,16,21)(13,24)(17,23,30,28)(19,22)(20,25)(27,31,29,32);(1,3)(2,7,10,11)(4,16)(5,19,20,13)(6,14)(8,21,15,26)(9,24)(12,18)(17,27,30,29)(22,25)(23,32)(28,31);(1,4)(2,8)(3,12,14,16)(5,17)(6,18)(7,21,11,26)(9,23)(10,15)(13,27,19,29)(20,30)(22,31,24,32)(25,28);(1,5)(2,9)(3,13)(4,17)(6,20)(7,22)(8,23)(10,25)(11,24)(12,27)(14,19)(15,28)(16,29)(18,30)(21,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237760, #72237763, #72237768, …
maximal normal subgroup of	#72237760, #72237763, #72237768, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357155);