#6436632

order	512
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3378 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)(20,28)(22,25)(23,30)(26,32);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(3,9)(6,12)(10,15)(13,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(2,7)(6,12)(8,14)(13,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,13,12,16)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249917, #72268350, #72268363, …
maximal normal subgroup of	#72249917, #72268350, #72268363, …
maximal quotient of	#72249970

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343828);