#6476522

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470362
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36624	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2058 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305296, #72305320
maximal normal subgroup of	#72305296, #72305320
maximal quotient of	#72306013, #72316046

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6383718);