#356108

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2209 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)(5,6,10,17,11,18,12,16)(7,19,21,13,22,31,23,32)(8,14,25,33,20,34,24,30)(9,15,28,35,27,36,26,29);(1,2)(3,4)(5,7,12,23,11,22,10,21)(6,13,17,31,18,32,16,19)(8,9,25,28,20,27,24,26)(14,29,30,36,34,35,33,15);(5,8)(6,14)(7,9)(10,24)(11,20)(12,25)(13,29)(15,19)(16,33)(17,30)(18,34)(21,26)(22,27)(23,28)(31,36)(32,35);(2,4)(5,9,11,27)(6,15,18,36)(7,20,22,8)(10,28,12,26)(13,30,32,33)(14,19,34,31)(16,29,17,35)(21,24,23,25)

Polycyclic presentation:

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

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, 263304);