#180290

order	512
nilpotency class	3
isoclinism	isoclinic class of #180005
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2733	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,d^b=dgi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

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, 87486);