#6444422

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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 ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2967 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)(9,10,14,19,16,20,15,12)(11,21,24,17,22,23,13,18);(5,6)(7,8)(9,11,16,22)(10,17,20,18)(12,23,19,21)(13,15,24,14);(1,2,3,4)(6,8)(9,12,15,20,16,19,14,10)(11,18,13,23,22,17,24,21);(6,8)(9,13,15,22,16,24,14,11)(10,18,12,23,20,17,19,21);(9,14,16,15)(10,19,20,12)(11,13,22,24)(17,21,18,23)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315231, #72315252, #72315255
maximal normal subgroup of	#72315231, #72315252, #72315255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351618);