#194278

order	512
nilpotency class	3
isoclinism	isoclinic class of #18866
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101474);