#127249

order	512
nilpotency class	4
isoclinism	isoclinic class of #127249
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=ghi,f^2=g,g^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,c^b=ci,e^b=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240713, #72240742, #72240743, …
maximal normal subgroup of	#72240713, #72240742, #72240743, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34445);