#497319

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#550	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×D₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1168 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,d^c=di,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404515);