#194918

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1292 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)(5,6,9,14,10,12,8,13)(7,15,18,11,16,19,17,20)(21,22,27,30,26,23,25,29)(24,28,32,35,34,31,33,36);(1,2,3,4)(5,7)(6,11)(8,18)(9,17)(10,16)(12,20)(13,19)(14,15)(21,23,26,22)(24,31,34,28)(25,29,27,30)(32,35,33,36);(21,24)(22,28)(23,31)(25,32)(26,34)(27,33)(29,35)(30,36);(6,12)(7,16)(13,14)(17,18)(21,25,26,27)(22,29,23,30)(24,32,34,33)(28,35,31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102114);