#272639

order	512
nilpotency class	3
isoclinism	isoclinic class of #272539
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	#62128	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

858 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)(3,6)(4,11)(5,7)(8,13)(9,15)(10,14)(12,16)(17,18,22,24,20,25,21,26)(19,27,30,31,29,23,28,32);(2,7)(3,9)(4,12)(5,15)(8,16)(10,11)(17,19)(18,23)(20,29)(21,28)(22,30)(24,32)(25,27)(26,31);(1,3,6,2)(4,10,12,8)(5,7,15,9)(11,13,16,14)(18,24)(19,28)(21,22)(25,26)(29,30)(31,32);(1,4)(2,8)(3,10)(5,14)(6,12)(7,11)(9,16)(13,15)(18,25)(23,27)(24,26)(31,32)

Polycyclic presentation:

2^9|a^2=fhi,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=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, 179835);