#146795

order	512
nilpotency class	4
isoclinism	isoclinic class of #12750
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

966 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,d^b=dhi,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250371, #72250374, #72250380, …
maximal normal subgroup of	#72250371, #72250374, #72250380, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53991);