#484045

order	512
nilpotency class	4
isoclinism	isoclinic class of #484039
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#2184	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216771, #72216779, #72216788, …
maximal normal subgroup of	#72216771, #72216779, #72216788, …
maximal quotient of	#72225063, #72225120, #72227144, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391241);