#6451862

order	512
nilpotency class	3
isoclinism	isoclinic class of #23996
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2404
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂	C₂×#2404
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3922 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218347, #72233772, #72233773, …
maximal normal subgroup of	#72218347, #72233772, #72233773, …
maximal quotient of	#72237666, C₄×#23996, C₂×#6451862, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6359058);