#418465

order	512
nilpotency class	3
isoclinism	isoclinic class of #418464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60456	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1649 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325661);