#482574

order	512
nilpotency class	4
isoclinism	isoclinic class of #482357
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂×#392	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1159 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,f^d=fi,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251759, #72251767, #72251769, …
maximal normal subgroup of	#72251759, #72251767, #72251769, …
maximal quotient of	#72293071

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389770);