#265189

order	512
nilpotency class	3
isoclinism	isoclinic class of #265163
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	C₂×#3399	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

907 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)(4,5)(6,10)(7,9)(8,14)(12,15)(13,16)(19,27)(20,21)(23,31)(24,25)(28,32)(29,30);(1,2,7,4,6,10,5,9)(3,8,14,11,15,16,13,12)(17,18,22,26)(19,28,29,31)(20,25,21,24)(23,27,32,30);(1,3,6,15)(2,8,10,16)(4,11,9,12)(5,13,7,14)(17,19,22,29)(18,23,26,32)(20,30,21,27)(24,28,25,31);(1,4,7,10,6,9,5,2)(3,12,14,8,15,11,13,16)(19,29)(23,32)(27,30)(28,31)

Polycyclic presentation:

2^9|b^2=ei,c^2=i,d^2=eh,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,d^c=di,f^c=fi,g^c=ghi

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, 172385);