#7632471

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

11342 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)(9,10,16,18)(11,12,14,13)(15,17,23,24)(19,20,22,21);(5,6,7,8)(9,11,16,14)(10,13,18,12)(15,22,23,19)(17,20,24,21);(1,2)(3,4)(9,12,16,13)(10,11,18,14)(15,20,23,21)(17,19,24,22);(9,13,16,12)(10,11,18,14)(15,21,23,20)(17,19,24,22);(9,14,16,11)(10,13,18,12)(15,22,23,19)(17,21,24,20);(9,15)(10,17)(11,19)(12,20)(13,21)(14,22)(16,23)(18,24)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=i,e^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, 7539667);