#285175

order	512
nilpotency class	3
isoclinism	isoclinic class of #285115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

697 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=hi,e^2=h,f^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=fi,e^d=eh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192371);