#6444386

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
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	#34936	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2968 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)(9,10,12,17,14,16,13,18)(11,19,20,23,22,24,21,15);(5,6)(7,8)(9,11,14,22)(10,15,16,23)(12,21,13,20)(17,24,18,19);(1,2,3,4)(6,8)(9,10,12,17,14,16,13,18)(11,19,20,23,22,24,21,15);(6,8)(9,10,12,17,14,16,13,18)(11,19,20,23,22,24,21,15);(10,16)(15,23)(17,18)(19,24)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307365, #72307626, #72315252, …
maximal normal subgroup of	#72307365, #72307626, #72315252, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351582);