#6508680

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499837
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	C₂²×C₄	C₂⁵
prime radical	#62819	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3098 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268066, #72268071, #72268087
maximal normal subgroup of	#72268066, #72268071, #72268087

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415876);