#7514108

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

3217 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)(10,16)(11,19)(12,14)(13,18)(17,22)(20,24);(1,2)(3,4)(5,6)(7,8)(9,10,15,18)(11,20,19,17)(12,16,14,13)(21,24,23,22);(6,8)(9,11,15,19)(10,17,18,20)(12,23,14,21)(13,22,16,24);(2,4)(6,8)(9,12,15,14)(10,16,18,13)(11,21,19,23)(17,22,20,24);(9,13)(10,12)(11,22)(14,18)(15,16)(17,23)(19,24)(20,21)

Polycyclic presentation:

2^9|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=cg,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, 7421304);