#433298

order	512
nilpotency class	3
isoclinism	isoclinic class of #433298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15476
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

1210 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)(4,15)(5,19)(6,10)(8,18)(9,20)(12,21,16,23)(13,25,27,22)(17,24)(26,31,29,32)(28,30);(1,3)(2,7)(4,16,18,12)(5,13)(6,14)(8,23,15,21)(9,25)(10,11)(17,29,30,26)(19,22)(20,27)(24,31,28,32);(1,4)(2,8,10,15)(3,12)(5,17)(6,18)(7,21,11,23)(9,24,19,28)(13,26)(14,16)(20,30)(22,31,25,32)(27,29);(1,5)(2,9,10,19)(3,13)(4,17)(6,20)(7,22,11,25)(8,24,15,28)(12,26)(14,27)(16,29)(18,30)(21,31,23,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237790, #72237791, #72237792, …
maximal normal subgroup of	#72237790, #72237791, #72237792, …
maximal quotient of	#72265878

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340494);