#270644

order	512
nilpotency class	3
isoclinism	isoclinic class of #270619
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

839 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 177840);