#6353794

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353765
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60946	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1636 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,22,25)(19,21,24,31)(20,27,28,26)(23,29,32,30);(17,19,21,25,22,24,31,18)(20,26,29,32,28,27,30,23);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20)(18,23)(19,26)(21,29)(22,28)(24,27)(25,32)(30,31);(20,28)(23,32)(26,27)(29,30);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,22,31)(18,24,25,19)(20,29,28,30)(23,27,32,26)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306703, #72306706, #72306707, …
maximal normal subgroup of	#72306703, #72306706, #72306707, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260990);