#357151

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1079 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 264347);