#615502

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5814 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303271, #72303294, #72303306, …
maximal normal subgroup of	#72303271, #72303294, #72303306, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 522698);