#418493

order	512
nilpotency class	3
isoclinism	isoclinic class of #418492
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

1667 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=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, 325689);