#473443

order	512
nilpotency class	3
isoclinism	isoclinic class of #473443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16865
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

975 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)(3,7,13,21)(4,8)(5,17)(6,10)(9,18)(11,25,14,19)(12,27,29,20)(15,22)(16,23)(24,30)(26,32,28,31);(1,3)(2,7,10,21)(4,14,16,11)(5,12)(6,13)(8,19)(9,20,17,27)(15,28,30,26)(18,29)(22,31)(23,25)(24,32);(1,4)(2,8)(3,11)(5,15)(6,16)(7,19)(9,24)(10,23)(12,28)(13,14)(17,22)(18,30)(20,31)(21,25)(26,29)(27,32);(1,5)(2,9,10,17)(3,12)(4,15)(6,18)(7,20,21,27)(8,22,23,24)(11,26)(13,29)(14,28)(16,30)(19,31,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237732, #72237762, #72237765, …
maximal normal subgroup of	#72237732, #72237762, #72237765, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380639);