#194580

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
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	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1206 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)(7,13,14,20)(8,12)(9,11)(10,15,18,17)(16,19)(21,22,25,30,26,28,24,29)(23,31,34,27,32,36,33,35);(1,2,3,4)(8,16)(12,19)(13,20)(15,17)(21,23)(22,27)(24,34)(25,33)(26,32)(28,35)(29,36)(30,31);(5,7)(6,10)(8,17)(9,14)(11,18)(12,20)(13,19)(15,16)(22,28)(24,25)(27,35)(33,34);(6,11)(10,18)(12,19)(13,20)(23,32)(27,35)(31,36)(33,34)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dhi,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101776);