#6591502

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252417, #72252423, #72252425
maximal normal subgroup of	#72252417, #72252423, #72252425
maximal quotient of	#72267130, #72267147

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6498698);