#273458

order	512
nilpotency class	3
isoclinism	isoclinic class of #273435
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61293	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

849 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=ghi,e^d=ei,f^d=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180654);