#9787120

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

5719 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256306, #72262970, #72262996
maximal normal subgroup of	#72256306, #72262970, #72262996

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9694316);