#7633152

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

11221 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540348);