#6366940

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366870
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3543	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1536 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306988, #72306999, #72307016
maximal normal subgroup of	#72306988, #72306999, #72307016

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274136);