#274071

order	512
nilpotency class	3
isoclinism	isoclinic class of #273947
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	#61278	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

851 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=hi,e^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fhi,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, 181267);