#457682

order	512
nilpotency class	3
isoclinism	isoclinic class of #457682
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16299
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

987 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)(5,17,19,9)(6,10)(11,20)(12,27,29,21)(13,22)(14,30)(15,23,18,25)(16,24)(26,31,28,32);(1,3)(2,7)(4,14,16,11)(5,12)(6,13)(8,20,24,30)(9,21)(10,22)(15,28,18,26)(17,27)(19,29)(23,31,25,32);(1,4)(2,8)(3,11)(5,18,19,15)(6,16)(7,20)(9,25,17,23)(10,24)(12,28,29,26)(13,14)(21,32,27,31)(22,30);(1,5)(2,9)(3,12)(4,15)(6,19)(7,21)(8,23)(10,17)(11,26)(13,29)(14,28)(16,18)(20,31)(22,27)(24,25)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237789, #72237794, #72237796, …
maximal normal subgroup of	#72237789, #72237794, #72237796, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364878);