#7985176

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

6327 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342307, #72342321, #72342331
maximal normal subgroup of	#72342307, #72342321, #72342331

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7892372);