#7514136

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

3191 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^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, 7421332);