#194520

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1102 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)(7,14,15,17)(11,18,20,19)(21,22,26,28,27,30,25,29)(23,31,34,24,33,35,32,36);(1,2,3,4)(5,6)(7,11)(8,12)(9,16)(10,13)(14,19)(15,20)(17,18)(21,23)(22,24)(25,34)(26,32)(27,33)(28,31)(29,35)(30,36);(5,7)(6,11)(8,14)(9,17)(10,15)(12,19)(13,20)(16,18)(21,24)(22,23)(25,35)(26,31)(27,36)(28,32)(29,34)(30,33);(5,8,10,9)(6,12,13,16)(7,14,15,17)(11,19,20,18)(21,25,27,26)(22,28,30,29)(23,32,33,34)(24,35,36,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,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, 101716);