#208931

order	512
nilpotency class	3
isoclinism	isoclinic class of #18132
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

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, 116127);