#7744697

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

6659 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:

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

Polycyclic presentation:

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

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, 7651893);