#7697514

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

11197 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)(5,6,7,8)(9,10,15,13)(11,12,19,16)(14,17,22,21)(18,20,24,23);(5,6)(7,8)(9,11,15,19)(10,16,13,12)(14,18,22,24)(17,23,21,20);(2,4)(9,12,15,16)(10,11,13,19)(14,20,22,23)(17,18,21,24);(1,2)(3,4)(9,12)(10,11)(13,19)(14,20)(15,16)(17,18)(21,24)(22,23);(9,13,15,10)(11,12,19,16)(14,17,22,21)(18,23,24,20);(9,14,15,22)(10,17,13,21)(11,18,19,24)(12,20,16,23)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=i,f^2=i,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, 7604710);