#6355593

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355552
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60916	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306780, #72306837, #72306846
maximal normal subgroup of	#72306780, #72306837, #72306846

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262789);