#198776

order	512
nilpotency class	3
isoclinism	isoclinic class of #198534
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi

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, 105972);