#397219

order	512
nilpotency class	3
isoclinism	isoclinic class of #394382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1448 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304415);