#7698343

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

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

10841 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)(3,4)(6,8)(9,10)(11,19)(12,17)(13,18)(14,15)(16,20)(21,23)(22,24);(5,6)(7,8)(9,10,14,15)(11,16,20,19)(12,17,22,24)(13,18,23,21);(1,2,3,4)(10,15)(11,20)(17,24)(18,21);(1,2)(3,4)(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,22)(23,24);(9,12)(10,17)(11,21)(13,16)(14,22)(15,24)(18,20)(19,23);(9,13)(10,18)(11,17)(12,19)(14,23)(15,21)(16,22)(20,24)

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7605539);