#7804790

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

5471 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)(17,18,20,22)(19,21,23,24);(1,2)(3,7)(4,8)(5,6)(17,18)(19,21)(20,22)(23,24);(2,6)(4,7)(10,13)(14,15)(17,19,20,23)(18,21,22,24);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16);(1,3)(2,4)(5,8)(6,7)(9,11,12,16)(10,14,13,15);(1,4)(2,3)(5,7)(6,8)

Polycyclic presentation:

2^9|a^2=g,c^2=g,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,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, 7711986);