#273198

order	512
nilpotency class	3
isoclinism	isoclinic class of #273179
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180394);