#6348457

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348447
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#3549	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306724, #72306737, #72306749, …
maximal normal subgroup of	#72306724, #72306737, #72306749, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255653);