#127216

order	512
nilpotency class	4
isoclinism	isoclinic class of #127215
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄³

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

725 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)(4,16,18,7)(5,10)(6,12)(9,19,22,32)(11,20)(13,24)(14,21,17,30)(15,26)(23,28,27,31)(25,29);(2,7,10,9)(3,13)(8,23,24,30)(11,22,12,16)(14,28)(15,29)(17,31)(21,26,27,25);(1,3,6,15)(2,8,12,26)(4,17,18,14)(5,13,20,29)(7,21,16,30)(9,27,22,23)(10,24,11,25)(19,31,32,28)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,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	#72237048, #72238398, #72238401, …
maximal normal subgroup of	#72237048, #72238398, #72238401, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34412);