#6628883

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628883
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2846
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

17022 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72233783, #72241001, #72241002
maximal subgroup of	#72215726, #72215784, #72215925, …
maximal normal subgroup of	#72215784, #72215925, #72219689, …
maximal quotient of	#72226680, #72226690, #72226694, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536079);