#7630836

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

11050 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)(6,7)(9,10)(11,20)(12,18)(13,15)(14,17)(16,19)(21,24)(22,23);(5,6,8,7)(9,11,16,14)(10,17,19,20)(12,23,13,21)(15,22,18,24);(1,2)(3,4)(5,7,8,6)(9,12,16,13)(10,18,19,15)(11,21,14,23)(17,22,20,24);(9,13,16,12)(10,15,19,18)(11,21,14,23)(17,22,20,24);(9,14,16,11)(10,17,19,20)(12,23,13,21)(15,24,18,22);(9,15)(10,12)(11,22)(13,19)(14,24)(16,18)(17,23)(20,21)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538032);