#8898085

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

5608 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=g,e^2=g,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, #72262874, #72262877
maximal normal subgroup of	#72255696, #72262874, #72262877

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8805281);