#194602

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6)(7,14,16,19)(8,17)(9,12)(10,13)(11,15,20,18)(21,22,27,29,26,31,25,30)(23,32,33,28,35,36,34,24);(1,2,3,4)(5,6)(7,11)(8,12)(9,17)(10,13)(14,18)(15,19)(16,20)(21,23)(22,28)(24,31)(25,33)(26,35)(27,34)(29,32)(30,36);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17)(21,24,26,28)(22,23,31,35)(25,32,27,36)(29,34,30,33);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(21,25,26,27)(22,29,31,30)(23,33,35,34)(24,36,28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,f^a=fh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,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, 101798);