#4315028

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2669	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3672 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)(11,13)(12,15)(14,16)(17,18)(19,23)(20,22)(21,24);(9,10,13,11)(12,15,16,14)(17,18,20,22)(19,21,24,23);(5,6,7,8)(10,11)(14,15)(19,24)(21,23);(1,2)(3,4)(9,11,13,10)(12,15,16,14)(17,19)(18,21)(20,24)(22,23);(9,12)(10,14)(11,15)(13,16)(17,18)(19,21)(20,22)(23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327758, #72327794, #72327803, …
maximal normal subgroup of	#72327758, #72327794, #72327803, …
maximal quotient of	#72333298

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4222224);