#6368580

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

1591 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)(19,24)(20,25)(21,23)(22,27)(26,29)(28,31)(30,32);(1,3)(2,7)(4,10)(5,11)(6,13)(8,16)(9,14)(12,15)(17,19,20,27)(18,22,25,24)(21,28,29,32)(23,30,26,31);(1,4,6,2)(3,10,13,7)(5,14,15,8)(9,12,16,11)(18,23)(21,29)(22,30)(24,31)(25,26)(28,32);(3,11)(5,15)(7,16)(8,14)(9,10)(12,13)(17,19)(18,24)(20,27)(21,28)(22,25)(23,31)(26,30)(29,32);(2,4)(7,10)(8,14)(9,16)(19,27)(22,24)(28,32)(30,31)

Polycyclic presentation:

2^9|b^2=i,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	#72306619, #72306650, #72306681
maximal normal subgroup of	#72306619, #72306650, #72306681

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275776);