#362830

order	512
nilpotency class	3
isoclinism	isoclinic class of #214656
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1481 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270026);