#210065

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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×#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1041 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,e^b=eh,f^b=fi,d^c=di,e^c=ei,f^c=fi

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, 117261);