#678228

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3374 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257907, #72257920, #72257925
maximal normal subgroup of	#72257907, #72257920, #72257925
maximal quotient of	#72308892

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 585424);