#466635

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

927 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=fi,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, 373831);