#120775

order	512
nilpotency class	3
isoclinism	isoclinic class of #10549
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

503 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:

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

Polycyclic presentation:

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

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, 27971);