#8898965

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255696, #72263021, #72263024
maximal normal subgroup of	#72255696, #72263021, #72263024

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8806161);