#423627

order	512
nilpotency class	3
isoclinism	isoclinic class of #422233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1532 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)(7,17,20,29)(8,22,26,13)(9,15,10,14)(12,21,28,18)(19,30,33,35)(23,27,34,36)(24,31,25,32);(1,2)(3,4)(5,7)(6,12)(8,23)(9,21)(10,18)(11,20)(13,30)(14,29)(15,17)(16,28)(19,24)(22,35)(25,33)(26,34)(27,31)(32,36);(2,4)(7,18,20,21)(12,17,28,29)(19,23,33,34)(27,30,36,35);(5,8,9,24,11,26,10,25)(6,13,14,31,16,22,15,32)(7,19,18,23,20,33,21,34)(12,27,17,30,28,36,29,35)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,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, 330823);