#265133

order	512
nilpotency class	3
isoclinism	isoclinic class of #265099
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10176	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

895 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)(3,11)(4,6)(5,9)(7,10)(8,13)(12,15)(14,16)(17,18)(19,26)(20,24)(21,23)(22,28)(25,30)(27,32)(29,31);(1,2,6,9,7,10,5,4)(3,12,15,11,14,8,13,16)(17,19,20,28)(18,22,24,26)(21,27,30,29)(23,31,25,32);(1,3)(2,8)(4,11)(5,13)(6,15)(7,14)(9,16)(10,12)(18,23)(19,27)(21,30)(24,25)(28,29)(31,32);(1,4,6,2,7,9,5,10)(3,11,15,8,14,16,13,12)(18,24)(22,26)(23,25)(31,32)

Polycyclic presentation:

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

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, 172329);