#7633161

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

11024 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)(5,6,7,8)(9,10,15,19)(11,16,14,18)(12,13,17,24)(20,21,22,23);(5,6)(7,8)(11,14)(16,18)(20,22)(21,23);(1,2,3,4)(9,11,15,14)(10,16,19,18)(12,20,17,22)(13,21,24,23);(9,12)(10,13)(11,20)(14,22)(15,17)(16,21)(18,23)(19,24);(9,13)(10,17)(11,21)(12,19)(14,23)(15,24)(16,22)(18,20);(9,14,15,11)(10,18,19,16)(12,22,17,20)(13,23,24,21)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,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, 7540357);