#6574175

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2777 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284913, #72285009
maximal normal subgroup of	#72284913, #72285009

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481371);