#7659584

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

6200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304327, #72304359, #72304372
maximal normal subgroup of	#72304327, #72304359, #72304372

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566780);