#479945

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄×C₈	C₂⁴
commutator, socle	C₂²	C₂²×C₄×C₈
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(18,22)(23,28)(24,29)(30,32);(17,18,19,23,20,24,25,30,21,22,26,28,27,29,31,32);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(18,22)(23,28)(24,29)(30,32);(3,9)(6,12)(10,15)(13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	M₃₂×#131, #72231856, #72231893, …
maximal normal subgroup of	M₃₂×#131, #72231856, #72231893, …
maximal quotient of	#72322698

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387141);