#6434129

order	512
nilpotency class	3
isoclinism	isoclinic class of #30056
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2631
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2631
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1945 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)(3,14)(4,10)(5,22)(6,12)(7,24)(8,13)(9,17)(11,15)(16,31)(18,27)(19,26)(20,29)(21,30)(23,32)(25,28);(1,3)(2,9)(4,19)(5,16)(6,17)(7,25)(8,18)(10,11)(12,27)(13,28)(14,24)(15,20)(21,23)(22,29)(26,30)(31,32);(1,4)(2,10)(3,15)(5,17)(6,20)(7,21)(8,16)(9,26)(11,27)(12,29)(13,31)(14,32)(18,23)(19,25)(22,28)(24,30);(1,5,7,15,8,19,6,23)(2,11,12,22,13,32,24,26)(3,16,25,20,18,4,17,21)(9,10,27,29,28,31,14,30);(1,6,8,7)(2,12,13,24)(3,17,18,25)(4,20,16,21)(5,23,19,15)(9,27,28,14)(10,29,31,30)(11,22,32,26)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279606, #72284483, #72287267, …
maximal normal subgroup of	#72279606, #72284483, #72287267, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341325);