#364584

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	D₄×C₂²⋊C₄
prime radical	#60366	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2374 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)(5,6,11,16)(7,17,21,28)(8,22)(9,15,10,14)(12,20,29,19)(13,24)(18,27)(23,30)(25,32)(26,31)(33,35)(34,36);(5,7,11,21)(6,12,16,29)(8,23)(9,20,10,19)(13,30)(14,28,15,17)(18,25)(22,27)(24,34)(26,33)(31,36)(32,35);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,22)(15,31)(16,32)(17,30)(19,33)(20,23)(21,34)(28,35)(29,36);(5,7)(6,12)(8,18)(9,20)(10,19)(11,21)(13,27)(14,28)(15,17)(16,29)(22,35)(23,24)(25,33)(26,34)(30,31)(32,36)

Polycyclic presentation:

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

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, 271780);