#482551

order	512
nilpotency class	4
isoclinism	isoclinic class of #482357
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁶	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1426 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)(3,13,15,27)(4,17,18,10)(5,19,21,11)(6,9)(7,12)(8,14,24,29)(16,28,31,32)(20,25,30,23)(22,26);(1,3,7,16)(2,8,12,25)(4,14,18,30)(5,20,21,29)(6,15,22,31)(9,23,26,24)(10,13,17,28)(11,27,19,32);(2,9)(8,23)(10,11)(12,26)(13,27)(17,19)(24,25)(28,32);(1,4,6,5)(2,10,9,11)(3,14,15,20)(7,18,22,21)(8,13,23,27)(12,17,26,19)(16,30,31,29)(24,32,25,28)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,e^2=h,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	#72241080, #72241091, #72241093, …
maximal normal subgroup of	#72241080, #72241091, #72241093, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389747);