#270717

order	512
nilpotency class	3
isoclinism	isoclinic class of #270683
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	#61963	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1008 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 177913);