#495192

order	512
nilpotency class	4
isoclinism	isoclinic class of #480482
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1230 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=f,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,d^c=di,g^e=gi

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, 402388);