#353107

order	512
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₄⁴ · Schur multiplier (H₂) = C₂×C₄⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂	C₄⁴
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

(13,14,16,20,17,19,23,26)(15,18,22,25,21,24,27,28);(1,2,3,4)(13,15)(14,18)(16,22)(17,21)(19,24)(20,25)(23,27)(26,28);(5,6,7,8)(15,21)(18,24)(22,27)(25,28);(9,10,11,12)(14,19)(18,24)(20,26)(25,28)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=h,e^2=i,d^a=di,c^b=ci

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239194, #72257731, #72257825
maximal normal subgroup of	#72239194, #72257731, #72257825
maximal quotient of	#72239195

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260303);