#271906

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	#60596	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

940 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=hi,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, 179102);