#464648

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1519 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)(7,15)(8,12)(9,22)(10,14)(11,17)(13,23)(16,30)(18,31)(19,28)(20,27)(21,29)(24,32)(25,35)(26,33)(34,36);(5,7)(6,11,14,25)(8,16)(9,17)(10,18)(12,26,29,24)(13,15,22,31)(19,32)(20,33)(21,34)(23,35)(27,36,28,30);(2,4)(5,8,10,21)(6,12,14,29)(7,16,18,34)(9,19,23,20)(11,24,25,26)(13,27,22,28)(15,30,31,36)(17,32,35,33);(8,19)(9,23)(12,27)(13,22)(15,31)(16,32)(17,35)(20,21)(24,30)(26,36)(28,29)(33,34)

Polycyclic presentation:

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

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, 371844);