#481961

order	512
nilpotency class	4
isoclinism	isoclinic class of #481819
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂×#2635	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1213 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)(3,9)(4,16,18,7)(5,8)(6,12)(10,17,21,31)(11,19)(13,26)(14,23,27,20)(15,22)(24,30,32,29)(25,28);(2,7)(3,13)(4,17)(8,10)(9,23)(11,21)(12,16)(14,29)(15,28)(18,31)(20,26)(22,32)(24,25)(27,30);(2,8)(7,10)(9,22)(11,12)(16,21)(20,24)(23,32)(25,26);(1,3,5,15)(2,9,8,22)(4,14,17,30)(6,13,19,28)(7,20,10,24)(11,25,12,26)(16,23,21,32)(18,27,31,29)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233258, #72233309, #72233311, …
maximal normal subgroup of	#72233258, #72233309, #72233311, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389157);