#234443

order	512
nilpotency class	3
isoclinism	isoclinic class of #234438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#37536	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1453 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,d^c=dhi,e^c=eh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141639);