#357150

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
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

1069 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)(5,6,11,16)(7,17,22,32)(8,23,27,9)(10,24,12,15)(13,21,33,20)(14,25,29,26)(18,19,35,30)(28,36,34,31);(5,7,12,20,11,22,10,21)(6,13,15,32,16,33,24,17)(8,18,26,36,27,35,25,31)(9,28,29,19,23,34,14,30);(1,2,4,3)(5,8,11,27)(6,14,16,29)(7,18,22,35)(9,24,23,15)(10,25,12,26)(13,30,33,19)(17,34,32,28)(20,36,21,31);(1,3,4,2)(5,9,10,14,11,23,12,29)(6,8,24,25,16,27,15,26)(7,19,21,28,22,30,20,34)(13,31,17,35,33,36,32,18)

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,f^c=fi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264346);