#417669

order	512
nilpotency class	3
isoclinism	isoclinic class of #417669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15257
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2339 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)(3,13)(4,9)(5,21)(6,14)(7,11)(8,12)(10,23)(15,31)(16,20)(17,19)(18,32)(22,27)(24,29)(25,28)(26,30);(1,3)(2,4)(5,22)(6,16)(7,9)(8,17)(10,27)(11,19)(12,20)(13,14)(15,23)(18,21)(24,26)(25,32)(28,30)(29,31);(1,4)(2,9)(3,14)(5,18)(6,13)(7,19)(8,20)(10,26)(11,17)(12,16)(15,21)(22,28)(23,27)(24,31)(25,30)(29,32);(1,5)(2,10)(3,15)(4,18)(6,25)(7,23)(8,24)(9,26)(11,29)(12,28)(13,30)(14,21)(16,22)(17,32)(19,27)(20,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213603, #72213913, #72215968, …
maximal normal subgroup of	#72213603, #72213913, #72215968, …
maximal quotient of	C₂×#417669

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324865);