#494355

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	C₂²×#41	C₂³
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1134 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:

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

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,f^b=fhi,d^c=di,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, 401551);