#6342516

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄²	C₂⁵
commutator, socle	C₂²	C₂³×C₄²
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2227 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,g^2=i,b^a=bh,c^b=ci,e^d=eh

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319723, #72319828, #72320015, …
maximal normal subgroup of	#72319723, #72319828, #72320015, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249712);