#194288

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

1206 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)(5,6,10,16,11,14,9,15)(7,17,19,12,8,13,18,20)(21,22,25,27,26,30,24,29)(23,31,33,28,34,35,32,36);(1,2,3,4)(5,7,11,8)(6,12,14,20)(9,19,10,18)(13,16,17,15)(22,27)(24,25)(28,31)(29,30)(32,33)(35,36);(5,8,11,7)(6,13,14,17)(9,19,10,18)(12,16,20,15)(21,23,26,34)(22,28,30,36)(24,33,25,32)(27,31,29,35);(6,14)(7,8)(15,16)(18,19)(21,24,26,25)(22,29,30,27)(23,32,34,33)(28,31,36,35)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=h,d^2=i,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, 101484);