#362647

order	512
nilpotency class	3
isoclinism	isoclinic class of #25446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1298 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:

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

Polycyclic presentation:

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

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, 269843);