#367941

order	512
nilpotency class	3
isoclinism	isoclinic class of #25446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1129 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,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, 275137);