#197566

order	512
nilpotency class	3
isoclinism	isoclinic class of #197354
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	#37440	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1426 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)(17,18,20,24,21,25,28,22)(19,23,27,31,26,30,32,29);(1,3,5,10)(2,6)(4,9,12,15)(7,13)(8,14)(11,16)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(17,19)(18,23)(20,27)(21,26)(22,29)(24,31)(25,30)(28,32);(19,26)(23,30)(27,32)(29,31)

Polycyclic presentation:

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

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, 104762);