#454048

order	512
nilpotency class	3
isoclinism	isoclinic class of #454045
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#35088	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|c^2=i,d^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,d^c=de

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361244);