#7737414

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	C₂²×#127	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5750 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)(3,8)(4,7)(5,6)(17,18,20,22)(19,23,24,21);(1,2)(3,4)(5,6)(7,8)(9,10)(11,16)(12,13)(14,15)(17,19,20,24)(18,21,22,23);(2,6)(3,7)(10,15)(11,12);(9,11,14,12)(10,16,15,13);(1,3,5,7)(2,4,6,8)(9,12,14,11)(10,13,15,16);(1,4)(2,7)(3,6)(5,8)(9,13)(10,11)(12,15)(14,16)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7644610);