#6574171

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

2697 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)(13,17,23,20)(15,18,24,22);(1,2,3,4)(9,11,14,21)(10,16,12,19)(13,15,23,24)(17,22,20,18);(6,8)(9,12,14,10)(11,20,21,17)(13,16,23,19)(15,18,24,22);(9,13)(10,17)(11,15)(12,20)(14,23)(16,22)(18,19)(21,24);(11,13)(15,24)(16,20)(17,19)(18,22)(21,23)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^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	#72263942, #72263980, #72263990
maximal normal subgroup of	#72263942, #72263980, #72263990
maximal quotient of	#72265198, #72308280

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481367);