#493674

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1190 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,d^c=di,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400870);