#456653

order	512
nilpotency class	3
isoclinism	isoclinic class of #456653
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16284
commutator, Frattini, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1606 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)(3,7,13,19)(4,8)(5,9)(6,10)(11,20,26,17)(12,18,28,31)(14,23)(15,22)(16,24)(21,29)(25,30,27,32);(1,3)(2,7,10,19)(4,11)(5,12)(6,13)(8,20,22,17)(9,18,24,31)(14,25)(15,26)(16,28)(21,32,23,30)(27,29);(1,4)(2,8)(3,11)(5,14)(6,15)(7,17)(9,23)(10,22)(12,27)(13,26)(16,29)(18,30)(19,20)(21,24)(25,28)(31,32);(1,5)(2,9)(3,12)(4,14)(6,16)(7,18)(8,21)(10,24)(11,25)(13,28)(15,29)(17,30)(19,31)(20,32)(22,23)(26,27)

Polycyclic presentation:

2^9|a^2=f,b^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,d^c=de,h^c=hi,g^d=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237732, #72237746, #72237759, …
maximal normal subgroup of	#72237732, #72237746, #72237759, …
maximal quotient of	#72265867

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363849);