#6469080

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	#34963
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 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)(3,11)(4,20)(5,12)(6,14)(7,13)(8,15)(9,18)(10,17)(16,29)(19,31)(21,27)(22,26)(23,32)(24,28)(25,30);(1,3)(2,9)(4,16)(5,17)(6,24)(7,18)(8,19)(10,29)(11,14)(12,27)(13,31)(15,28)(20,32)(21,22)(23,25)(26,30);(1,4,7,17,8,21,6,25)(2,10,14,20,15,30,13,27)(3,16,18,5,19,22,24,23)(9,26,11,12,28,29,31,32);(1,3)(2,11)(4,16)(5,17)(6,24)(7,18)(8,19)(9,13)(10,12)(14,28)(15,31)(20,29)(21,22)(23,25)(26,27)(30,32);(1,5,7,22,8,23,6,16)(2,12,13,26,15,32,14,29)(3,17,18,21,19,25,24,4)(9,27,31,30,28,20,11,10)

Polycyclic presentation:

2^9|c^2=g,e^2=gh,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,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213945, #72231703, #72231715, …
maximal normal subgroup of	#72213945, #72231703, #72231715, …
maximal quotient of	#72224916, #72239887, #72257156, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376276);