#271947

order	512
nilpotency class	3
isoclinism	isoclinic class of #271899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60431	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1071 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179143);