#612502

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4537 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)(9,10,13,16)(11,15,12,14)(17,18)(19,23)(20,22)(21,24);(6,8)(9,11,13,12)(10,14,16,15)(17,18,20,22)(19,21,24,23);(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24);(1,2)(3,4)(9,12,13,11)(10,15,16,14);(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=g,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	#72304002, #72304108, #72304120, …
maximal normal subgroup of	#72304002, #72304108, #72304120, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 519698);