#235078

order	512
nilpotency class	3
isoclinism	isoclinic class of #235075
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,e^c=ei,f^c=fh

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, 142274);