#494352

order	512
nilpotency class	4
isoclinism	isoclinic class of #33551
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1185 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,b^a=be,c^a=cf,h^a=hi,d^b=di,f^b=fhi,e^c=eh,f^e=fi

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, 401548);