#428192

order	512
nilpotency class	3
isoclinism	isoclinic class of #417903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

955 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=ef,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,g^c=gi

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, 335388);