#268041

order	512
nilpotency class	3
isoclinism	isoclinic class of #267947
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61581	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1120 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)(3,7,4,6)(5,8)(9,10,13,18,15,20,14,19)(11,16,22,31,24,33,23,32)(12,25,29,17,28,36,27,35)(21,37,26,30,39,34,38,40);(1,3,5,4)(2,6,8,7)(9,11,15,24)(10,16,20,33)(12,26,28,38)(13,22,14,23)(17,34,35,37)(18,31,19,32)(21,29,39,27)(25,30,36,40);(1,2)(3,6)(4,7)(5,8)(9,12)(10,17)(11,21)(13,27)(14,29)(15,28)(16,30)(18,25)(19,36)(20,35)(22,38)(23,26)(24,39)(31,37)(32,34)(33,40);(1,4,5,3)(2,7,8,6)(9,13,15,14)(10,18,20,19)(11,22,24,23)(12,27,28,29)(16,31,33,32)(17,25,35,36)(21,38,39,26)(30,37,40,34)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi,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, 175237);