#608083

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14757	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262120, #72281347, #72281401, …
maximal normal subgroup of	#72262120, #72281347, #72281401, …
maximal quotient of	#72298465

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 515279);