#6444373

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	#35565	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2901 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)(9,10,15,19,14,17,13,11)(12,20,23,18,21,16,22,24);(5,6)(7,8)(9,11,14,19)(10,15,17,13)(12,20,21,16)(18,23,24,22);(1,2,3,4)(6,8)(9,12,15,23,14,21,13,22)(10,16,19,24,17,20,11,18);(6,8)(9,10,13,11,14,17,15,19)(12,16,22,18,21,20,23,24);(12,21)(16,20)(18,24)(22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315220, #72315231, #72315245
maximal normal subgroup of	#72315220, #72315231, #72315245
maximal quotient of	#72315780

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351569);