#268094

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	C₂×#548	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,6,8,4,3,5,7)(9,10,13,18,15,20,14,19)(11,16,22,31,23,32,29,36)(12,24,26,35,28,34,27,17)(21,33,25,39,37,40,38,30);(5,6)(7,8)(9,11,15,23)(10,16,20,32)(12,25,28,38)(13,22,14,29)(17,33,35,40)(18,31,19,36)(21,26,37,27)(24,39,34,30);(1,3,4,2)(5,7,6,8)(9,12,15,28)(10,17,20,35)(11,21,23,37)(13,27,14,26)(16,30,32,39)(18,34,19,24)(22,38,29,25)(31,40,36,33);(5,6)(7,8)

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^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, 175290);