#469774

order	512
nilpotency class	3
isoclinism	isoclinic class of #464719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

1039 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)(7,12,19,28)(8,21,23,32)(9,15,10,14)(13,24,31,25)(17,29,20,26)(18,34,22,36)(27,33,30,35);(7,17,19,20)(12,26,28,29)(18,33,22,35)(27,36,30,34);(5,7)(6,12)(8,22,23,18)(9,20)(10,17)(11,19)(13,30,31,27)(14,29)(15,26)(16,28)(21,34,32,36)(24,33,25,35);(2,4)(5,8,9,24,11,23,10,25)(6,13,14,32,16,31,15,21)(7,18,20,35,19,22,17,33)(12,27,29,34,28,30,26,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376970);