#148005

order	512
nilpotency class	4
isoclinism	isoclinic class of #137768
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

837 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,f^b=fi,f^c=fi,g^c=gh,g^e=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55201);