#194194

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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1208 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)(5,6,9,13,10,14,8,12)(7,15,16,11,18,19,17,20)(21,22,26,30,27,23,25,29)(24,32,34,28,35,36,33,31);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,19)(13,15)(14,20)(21,23,27,22)(24,31,35,28)(25,29,26,30)(32,34,36,33);(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,19,20,15)(21,24,27,35)(22,28,23,31)(25,34,26,33)(29,36,30,32);(5,8,10,9)(6,12,14,13)(7,17,18,16)(11,15,20,19)(22,23)(28,31)(29,30)(32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,d^b=dh,e^b=eh,f^b=fi,g^b=gh,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, 101390);