#6436635

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3207 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318366
maximal normal subgroup of	#72318366

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343831);