#195178

order	512
nilpotency class	3
isoclinism	isoclinic class of #18224
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1302 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fhi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102374);