#6437584

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2403
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2403
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6011 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)(3,10)(4,18)(5,11)(6,12)(7,13)(8,16)(9,19)(14,22)(15,23)(17,28)(20,27)(21,26)(24,30)(25,29)(31,32);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,24)(10,11)(12,28)(13,23)(18,25)(19,29)(20,31)(21,30)(22,27)(26,32);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,25)(11,26)(12,18)(13,27)(15,29)(16,31)(17,30)(23,24)(28,32);(1,3)(2,10)(4,14)(5,15)(6,16)(7,17)(8,12)(9,25)(11,23)(13,28)(18,22)(19,29)(20,31)(21,30)(24,26)(27,32);(1,5,7,6)(2,11,13,12)(3,15,17,16)(4,19,21,20)(8,10,23,28)(9,26,27,18)(14,29,30,31)(22,25,24,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214632, #72215784, #72215901, …
maximal normal subgroup of	#72214632, #72215784, #72215901, …
maximal quotient of	C₄×#23872, #72256022, #72260459, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344780);