#127225

order	512
nilpotency class	4
isoclinism	isoclinic class of #127217
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×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

705 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gh,e^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cgi,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	#72251677, #72251721, #72251722, …
maximal normal subgroup of	#72251677, #72251721, #72251722, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34421);