#497277

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₂²⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1083 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)(3,4)(5,6,10,15,9,16,19,32,11,17,25,29,28,14,34,36)(7,18,21,13,22,31,26,33,23,35,8,24,20,12,27,30);(5,7)(6,12)(8,25)(9,22)(10,21)(11,23)(13,32)(14,35)(15,30)(16,18)(17,31)(19,26)(20,28)(24,36)(27,34)(29,33);(2,4);(5,8)(6,13)(7,19)(9,27)(10,20)(11,21)(12,29)(14,30)(15,31)(16,33)(17,24)(18,36)(22,25)(23,34)(26,28)(32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404473);