#127232

order	512
nilpotency class	4
isoclinism	isoclinic class of #127215
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,d^2=gh,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cg,e^b=eh,d^c=di,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234699, #72234704, #72234705, …
maximal normal subgroup of	#72234699, #72234704, #72234705, …
maximal quotient of	C₂×#127232

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34428);