#6358387

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈²
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂×D₄²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1752 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)(3,11)(4,6)(5,9)(7,10)(8,12)(13,15)(14,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,29)(27,31)(30,32);(1,3)(2,8)(4,11)(5,12)(6,14)(7,13)(9,15)(10,16)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,4)(2,5)(3,8)(6,10)(7,9)(11,14)(12,15)(13,16);(1,5,7,6)(2,9,10,4)(3,12,13,14)(8,15,16,11)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287660, #72287674, #72287679
maximal normal subgroup of	#72287660, #72287674, #72287679

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265583);