#208963

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1138 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)(5,6,9,12)(7,13,14,20)(8,16,18,11)(10,17,19,15)(21,22,25,28)(23,26,30,33)(24,27,31,34)(29,35,36,32);(5,7,9,14)(6,10,12,19)(8,17,18,15)(11,20,16,13)(21,23)(22,26)(24,29)(25,30)(27,32)(28,33)(31,36)(34,35);(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,36)(33,35);(1,3,4,2)(7,14)(10,19)(13,20)(15,17)(22,28)(26,33)(27,34)(32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116159);