#617199

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72332114, #72332140, #72332155
maximal normal subgroup of	#72332114, #72332140, #72332155
maximal quotient of	#72332498

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 524395);