#268125

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	#10128	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

831 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,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, 175321);