#386044

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#2539	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=e,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,h^c=hi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293240);