#484324

order	512
nilpotency class	4
isoclinism	isoclinic class of #484162
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

953 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227975, #72227986
maximal normal subgroup of	#72227975

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391520);