#615500

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	C₂²×#525	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6062 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)(10,14)(11,12)(17,18)(19,23)(20,21)(22,24);(2,4)(6,8)(9,10)(11,15)(12,16)(13,14)(18,21)(22,23);(1,2)(3,4)(5,6)(7,8)(9,11,13,12)(10,15,14,16)(17,18)(19,22)(20,21)(23,24);(9,10)(11,15)(12,16)(13,14);(9,12,13,11)(10,16,14,15)(17,19,20,24)(18,22,21,23)

Polycyclic presentation:

2^9|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	#72261775, #72261777, #72261780, …
maximal normal subgroup of	#72261775, #72261777, #72261780, …
maximal quotient of	#72333096, #72336024

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 522696);