#105801

order	512
nilpotency class	3
isoclinism	isoclinic class of #105783
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

812 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18)(19,27)(20,31,32,24)(21,34)(22,25)(23,29)(26,35)(28,36,41,39)(30,37)(33,38,45,44)(40,43)(42,47,48,46);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19,21,30,22,29,35,43)(18,23,26,37,25,27,34,40)(20,28,33,42,32,41,45,48)(24,36,38,46,31,39,44,47);(3,10)(6,14)(9,15)(11,16)(17,20)(18,24)(19,28)(21,33)(22,32)(23,36)(25,31)(26,38)(27,39)(29,41)(30,42)(34,44)(35,45)(37,46)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,c^b=ch,d^b=di,f^c=fi

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, 12997);