#6368584

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368558
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1562 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)(4,9)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18)(19,24)(20,25)(21,23)(22,27)(26,29)(28,31)(30,32);(2,7)(4,12)(5,15)(8,14)(9,16)(10,11)(17,19,20,27)(18,22,25,24)(21,28,29,32)(23,30,26,31);(1,3,6,13)(2,8,10,12)(4,7,14,11)(5,9,15,16)(18,23)(21,29)(22,30)(24,31)(25,26)(28,32);(1,4,6,14)(2,9,10,16)(3,7,13,11)(5,8,15,12)(17,19)(18,24)(20,27)(21,28)(22,25)(23,31)(26,30)(29,32);(2,10)(3,13)(7,11)(9,16)(19,27)(22,24)(28,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306650, #72306678, #72306697
maximal normal subgroup of	#72306650, #72306678, #72306697

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275780);