#268124

order	512
nilpotency class	3
isoclinism	isoclinic class of #268043
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	#3109	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

794 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:

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

Polycyclic presentation:

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

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, 175320);