#272410

order	512
nilpotency class	3
isoclinism	isoclinic class of #272283
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	#19292	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

872 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)(3,13)(4,11)(5,8)(6,9)(7,12)(10,14)(15,16)(17,18,23,25,22,20,21,26)(19,27,31,32,30,24,29,28);(1,3,7,14)(2,8,12,16)(4,5,6,15)(9,13,11,10)(17,19)(18,24)(20,27)(21,29)(22,30)(23,31)(25,28)(26,32);(1,4,7,6)(2,9,12,11)(3,5,14,15)(8,13,16,10)(17,20,22,18)(19,28,30,32)(21,26,23,25)(24,29,27,31);(1,5,7,15)(2,10,12,13)(3,4,14,6)(8,11,16,9)(17,21,22,23)(18,25,20,26)(19,29,30,31)(24,28,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179606);