#469777

order	512
nilpotency class	3
isoclinism	isoclinic class of #469776
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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1045 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)(7,17)(8,13)(9,14)(10,26)(11,16)(12,19)(15,25)(18,33)(20,32)(21,29)(22,36)(23,31)(24,27)(28,35)(30,34);(5,7,10,19,11,20,25,9)(6,12,15,17,16,14,26,32)(8,18,22,35,23,21,34,24)(13,28,30,33,31,27,36,29);(5,8,11,23)(6,13,16,31)(7,18)(9,24)(10,22,25,34)(12,27)(14,28)(15,30,26,36)(17,29)(19,35)(20,21)(32,33);(2,4)(5,9,25,20,11,19,10,7)(6,14,15,32,16,12,26,17)(8,21,22,24,23,18,34,35)(13,29,36,27,31,33,30,28)

Polycyclic presentation:

2^9|b^2=e,c^2=fi,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,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, 376973);