#846054

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4206 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)(9,10,13,14)(11,16,12,15)(17,18,21,24)(19,22,23,20);(5,6,7,8)(10,14)(15,16)(17,18)(19,22)(20,23)(21,24);(5,6)(7,8)(9,11,13,12)(10,15,14,16)(17,19,21,23)(18,22,24,20);(1,2)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,23)(21,24);(9,12,13,11)(10,16,14,15)(17,20,21,22)(18,23,24,19)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257607, #72257902, #72257907, …
maximal normal subgroup of	#72257607, #72257902, #72257907, …
maximal quotient of	#72264782, #72335941

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 753250);