#7684748

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5060 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)(4,6)(5,7)(10,14)(15,16)(17,18)(19,22)(20,24)(21,23);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,14)(17,19,21,24)(18,22,23,20);(1,3,5,6)(2,4,7,8)(9,11,13,12)(10,15,14,16)(17,20)(18,19)(21,22)(23,24);(1,2)(3,8)(4,6)(5,7)(17,18)(19,22)(20,24)(21,23);(1,2)(3,4)(5,7)(6,8)(9,11,13,12)(10,15,14,16);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,16,14,15)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,e^2=g,f^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7591944);