#8828675

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

5363 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)(3,7)(9,10)(11,13)(12,15)(14,16)(17,18,21,19)(20,23,24,22);(2,5)(3,7)(9,10,14,16)(11,13,15,12)(18,19)(22,23);(1,2)(3,6)(4,5)(7,8)(9,11,14,15)(10,12,16,13)(17,19,21,18)(20,23,24,22);(2,5)(6,8)(9,12,14,13)(10,11,16,15)(17,20,21,24)(18,22,19,23);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,15)(14,16);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,16,11)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303495, #72305957
maximal normal subgroup of	#72303495, #72305957
maximal quotient of	#72303510, #72305958, #72305960

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8735871);