#194121

order	512
nilpotency class	3
isoclinism	isoclinic class of #18562
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

1270 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)(5,6,8,13,9,12,17,18)(7,11,15,19,14,20,16,10)(21,22)(23,29,31,36)(24,26)(25,28)(27,30,34,33)(32,35);(1,2,3,4)(5,7)(6,10)(8,16)(9,14)(11,18)(12,19)(13,20)(15,17)(22,26)(25,32)(27,34)(30,33);(5,7)(6,11)(8,15)(9,14)(10,18)(12,20)(13,19)(16,17)(21,23)(22,27)(24,31)(25,33)(26,34)(28,36)(29,35)(30,32);(6,12)(7,14)(13,18)(15,16)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101317);