#7514258

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(5,6)(7,8)(9,10,16,12)(11,19,21,24)(13,18,15,23)(14,22,20,17);(1,2)(3,4)(5,6)(7,8)(9,11,16,21)(10,17,12,22)(13,20,15,14)(18,24,23,19);(6,8)(9,12,16,10)(11,17,21,22)(13,18,15,23)(14,24,20,19);(2,4)(6,8)(9,13,16,15)(10,18,12,23)(11,20,21,14)(17,24,22,19);(9,14)(10,19)(11,13)(12,24)(15,21)(16,20)(17,18)(22,23)

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=ef,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, 7421454);