#7514322

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3146 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)(11,18,13,26)(12,23,25,17)(14,29,30,19)(16,28,32,22)(21,33,27,35)(24,34,40,38)(31,36,37,39);(1,2)(3,4)(5,6)(7,8)(9,11,15,13)(10,16,20,32)(12,21,25,27)(14,28,30,22)(17,31,23,37)(18,19,26,29)(24,39,40,36)(33,38,35,34);(6,8)(9,12,15,25)(10,17,20,23)(11,21,13,27)(14,24,30,40)(16,31,32,37)(18,35,26,33)(19,34,29,38)(22,36,28,39);(2,4)(6,8)(9,13,15,11)(10,18,20,26)(12,21,25,27)(14,28,30,22)(16,19,32,29)(17,33,23,35)(24,36,40,39)(31,38,37,34);(10,19)(14,30)(16,26)(17,34)(18,32)(20,29)(22,28)(23,38)(24,40)(31,33)(35,37)(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=dg,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421518);