#230055

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,e^b=eh,g^b=gh,d^c=di,f^c=fh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137251);