#8726581

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

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

6848 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233550, #72233564, #72233598
maximal normal subgroup of	#72233550, #72233564, #72233598

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633777);