#7712738

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

6246 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304364, #72304417, #72304470
maximal normal subgroup of	#72304364, #72304417, #72304470

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619934);