#106199

order	512
nilpotency class	3
isoclinism	isoclinic class of #106194
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

739 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=dh,e^b=eh,f^b=fi,d^c=dh,e^c=eh,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13395);