#388276

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1048 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=dei,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295472);