#423639

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1535 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,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,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, 330835);