#435781

order	512
nilpotency class	3
isoclinism	isoclinic class of #435781
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15538
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

1169 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)(3,7)(4,8)(5,16,18,9)(6,10)(11,25)(12,20,17,24)(13,21)(14,28,29,22)(15,23)(19,27)(26,32,30,31);(1,3)(2,7)(4,11)(5,17)(6,13)(8,19)(9,24)(10,21)(12,18)(14,26)(15,27)(16,20)(22,31)(23,25)(28,32)(29,30);(1,4)(2,8)(3,11)(5,14)(6,15)(7,19)(9,22)(10,23)(12,26)(13,27)(16,28)(17,30)(18,29)(20,31)(21,25)(24,32);(1,5)(2,9)(3,12,13,17)(4,14)(6,18)(7,20,21,24)(8,22)(10,16)(11,26,27,30)(15,29)(19,31,25,32)(23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237746, #72237760, #72237767, …
maximal normal subgroup of	#72237746, #72237760, #72237767, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 342977);