#123732

order	512
nilpotency class	3
isoclinism	isoclinic class of #9109
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,g^b=gi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30928);