#210148

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
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

945 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)(6,10)(8,15)(11,17)(12,16)(14,19)(18,20)(21,22,26,29)(23,30,32,36)(24,28,25,34)(27,31,35,33);(5,6,9,12)(7,11,13,18)(8,16,15,10)(14,20,19,17)(24,25)(28,34)(30,36)(31,33);(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,19)(16,20)(21,23,26,32)(22,27,29,35)(24,33,25,31)(28,36,34,30);(1,3,4,2)(7,13)(11,18)(14,19)(17,20)(21,24,26,25)(22,28,29,34)(23,31,32,33)(27,30,35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=g,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=dhi,e^c=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117344);