#100019

order	512
nilpotency class	3
isoclinism	isoclinic class of #9345
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

657 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=gh,e^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,f^b=fh,e^c=egh

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, 7215);