#370118

order	512
nilpotency class	3
isoclinism	isoclinic class of #24420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1264 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,c^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi

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, 277314);