#6357178

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357172
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60940	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264374);