#778480

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4568 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)(6,8)(9,10)(11,14)(12,16)(13,15)(19,23)(21,24);(9,11,13,16)(10,12,15,14)(17,18,20,22)(19,21,23,24);(5,6)(7,8)(9,12,13,14)(10,11,15,16)(17,18)(19,21)(20,22)(23,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,19,20,23)(18,21,22,24);(9,11)(10,14)(12,15)(13,16)

Polycyclic presentation:

2^9|b^2=fh,c^2=f,d^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264265, #72303388, #72332064, …
maximal normal subgroup of	#72264265, #72303388, #72332064, …
maximal quotient of	#72332695

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 685676);