#194289

order	512
nilpotency class	3
isoclinism	isoclinic class of #18866
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1156 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)(5,6,9,11,10,14,8,13)(7,12,15,18,17,20,16,19)(21,22,24,29,26,31,25,30)(23,32,34,28,33,36,35,27);(1,2,3,4)(6,11)(8,9)(12,18)(13,14)(15,16)(19,20)(21,23)(22,27)(24,35)(25,34)(26,33)(28,31)(29,36)(30,32);(5,7)(6,12)(8,16)(9,15)(10,17)(11,18)(13,19)(14,20)(21,23)(22,28)(24,35)(25,34)(26,33)(27,31)(29,32)(30,36);(5,8,10,9)(6,11,14,13)(7,15,17,16)(12,19,20,18)(23,33)(27,28)(32,36)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101485);