#503228

order	512
nilpotency class	4
isoclinism	isoclinic class of #503227
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25771
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^c=gh,h^c=hi,g^f=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236997, #72237519, #72237521, …
maximal normal subgroup of	#72236997, #72237519, #72237521, …
maximal quotient of	#72311953, #72312138, #72312173, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410424);