#108365

order	512
nilpotency class	3
isoclinism	isoclinic class of #108352
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ei

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, 15561);