#7779443

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
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5390 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304361, #72304411, #72304467
maximal normal subgroup of	#72304361, #72304411, #72304467

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7686639);