#6574399

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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
prime radical	C₂×#3273	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2719 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)(9,10,15,21)(11,16,22,17)(12,18,23,24)(13,20,14,19);(1,2,3,4)(5,6)(7,8)(9,11)(10,16)(12,14)(13,23)(15,22)(17,21)(18,19)(20,24);(6,8)(9,12,15,23)(10,17,21,16)(11,13,22,14)(18,20,24,19);(5,6)(7,8)(9,12)(10,18)(11,13)(14,22)(15,23)(16,20)(17,19)(21,24);(6,8)(9,13,15,14)(10,19,21,20)(11,12,22,23)(16,24,17,18)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342367, #72342369, #72342374
maximal normal subgroup of	#72342367, #72342369, #72342374

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481595);