#7514129

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36085	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3129 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,d^c=di,f^e=fi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421325);