#6368561

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

1596 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)(3,9)(4,6)(5,14)(7,12)(8,15)(10,11)(13,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,30)(27,29)(31,32);(1,3,6,13)(2,7,4,10)(5,11,15,12)(8,16,14,9)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(1,4,6,2)(3,10,13,7)(5,14,15,8)(9,12,16,11)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(3,11)(5,15)(7,16)(8,14)(9,10)(12,13)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(2,4)(7,10)(8,14)(9,16)(19,26)(22,30)(27,32)(29,31)

Polycyclic presentation:

2^9|b^2=h,c^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, #72306684, #72306697
maximal normal subgroup of	#72306650, #72306684, #72306697

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275757);