#493726

order	512
nilpotency class	4
isoclinism	isoclinic class of #480274
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×#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

1174 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)(5,6,10,16,11,17,8,14)(7,12,18,26,20,28,21,30)(9,15,22,31,23,13,24,29)(19,27,34,35,33,25,32,36);(1,2,3,4)(5,7)(6,12)(8,21)(9,19)(10,18)(11,20)(13,25)(14,30)(15,27)(16,26)(17,28)(22,34)(23,33)(24,32)(29,36)(31,35);(6,13)(9,23)(12,25)(14,29)(15,17)(16,31)(19,33)(22,24)(26,35)(27,28)(30,36)(32,34);(5,8,11,10)(6,14,17,16)(7,18,20,21)(9,22,23,24)(12,26,28,30)(13,29,15,31)(19,32,33,34)(25,35,27,36)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fi,g^c=ghi,g^f=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, 400922);