#535425

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3500 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331704, #72331708, #72331722
maximal normal subgroup of	#72331704, #72331708, #72331722
maximal quotient of	#72333319

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 442621);