#7504666

order	512
nilpotency class	3
isoclinism	isoclinic class of #31414
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2688
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2688
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1937 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,h^a=hi,c^b=ch,d^b=df,f^b=fi,g^b=gi,g^c=gi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279606, #72284483, #72287261, …
maximal normal subgroup of	#72279606, #72284483, #72287261, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411862);