#7487869

order	512
nilpotency class	3
isoclinism	isoclinic class of #7487864
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#52004
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1875 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)(3,12)(4,8)(5,9)(6,11)(7,13)(10,18)(14,25)(15,28)(16,23)(17,24)(19,31)(20,26)(21,27)(22,29)(30,32);(1,3)(2,7)(4,16)(5,19)(6,15)(8,12)(9,25)(10,28)(11,23)(13,18)(14,20)(17,30)(21,29)(22,26)(24,31)(27,32);(1,4,6,18)(2,8,11,10)(3,13,15,16)(5,20,21,17)(7,12,23,28)(9,24,27,26)(14,19,30,29)(22,25,31,32);(1,5,18,20,6,21,4,17)(2,9,10,26,11,27,8,24)(3,14,16,19,15,30,13,29)(7,22,12,25,23,31,28,32);(2,8,11,10)(3,15)(7,12,23,28)(9,24,27,26)(13,16)(14,30)(19,29)(22,25,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265007, #72279292, #72279336, …
maximal normal subgroup of	#72265007, #72279292, #72279336, …
maximal quotient of	#72265649, C₂×#7487869

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7395065);