#127221

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

630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18)(19,27)(20,31,32,24)(21,25)(22,26)(23,29,35,41)(28,39)(30,40)(33,44,45,37)(34,38)(36,43,46,48)(42,47);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19,21,30,22,28,34,42)(18,23,25,36,26,35,38,46)(20,29,33,43,32,41,45,48)(24,27,37,40,31,39,44,47);(3,10)(6,14)(9,15)(11,16)(19,28)(20,32)(24,31)(27,39)(30,42)(33,45)(37,44)(40,47)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gh,f^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	#72284078, #72284113, #72284183, …
maximal normal subgroup of	#72284078, #72284113, #72284183, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34417);