#482597

order	512
nilpotency class	4
isoclinism	isoclinic class of #482397
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#35692	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1149 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)(3,13,14,23)(4,17,19,7)(5,8)(6,12)(9,16,25,29)(10,18,22,31)(11,20)(15,24)(21,30)(26,28)(27,32);(2,7)(4,18)(8,10)(9,21)(11,22)(12,17)(13,26)(15,30)(19,31)(23,24)(25,27)(28,32);(2,8)(3,14)(7,10)(9,13)(11,12)(15,28)(16,29)(17,22)(21,26)(23,25)(24,27)(30,32);(1,3,5,16)(2,9,8,23)(4,15,18,30)(6,14,20,29)(7,21,10,24)(11,13,12,25)(17,27,22,26)(19,28,31,32)

Polycyclic presentation:

2^9|a^2=f,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,d^c=di,f^d=fi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338739
maximal normal subgroup of	#72338739

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389793);