#6461447

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442596
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2598
prime radical	#23414	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1851 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)(3,7)(4,8)(5,10)(6,9)(11,27)(12,18)(13,21)(14,22)(15,28)(16,24)(17,25)(19,26)(20,29)(23,30)(31,32);(1,3)(2,7)(4,15)(5,18)(6,13)(8,23)(9,21)(10,12)(11,17)(14,19)(16,30)(20,25)(22,26)(24,28)(27,32)(29,31);(1,4,5,17,6,16,19,31)(2,8,10,25,9,24,26,32)(3,11,14,15,13,29,18,30)(7,20,22,23,21,27,12,28);(1,2)(3,12,13,22)(4,8)(5,10)(6,9)(7,18,21,14)(11,28,29,23)(15,20,30,27)(16,24)(17,25)(19,26)(31,32);(2,9)(7,21)(8,24)(10,26)(12,22)(20,27)(23,28)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306892, #72307055, #72317757, …
maximal normal subgroup of	#72306892, #72307055, #72317757, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368643);