#368465

order	512
nilpotency class	3
isoclinism	isoclinic class of #24408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1369 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=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, 275661);