#495155

order	512
nilpotency class	4
isoclinism	isoclinic class of #480460
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1271 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)(7,15,19,29)(8,12)(9,22,16,13)(10,14)(11,18)(17,28,31,36)(20,26)(21,33,30,27)(23,25)(24,32)(34,35);(5,7,16,18,10,19,9,23)(6,11,22,29,14,25,13,15)(8,17,30,32,20,31,21,34)(12,24,33,36,26,35,27,28);(1,2,3,4)(8,20)(12,26)(17,31)(21,30)(24,35)(27,33)(28,36)(32,34);(5,8,10,20)(6,12,14,26)(7,17,19,31)(9,21,16,30)(11,24,25,35)(13,27,22,33)(15,28,29,36)(18,32,23,34)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gh,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, 402351);