#7689716

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312217, #72342315, #72342325
maximal normal subgroup of	#72312217, #72342315, #72342325

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596912);