#237611

order	512
nilpotency class	3
isoclinism	isoclinic class of #237595
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1508 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281923, #72282522
maximal normal subgroup of	#72281923, #72282522
maximal quotient of	#72283329

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144807);