#197723

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1178 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ch,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104919);