#498170

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2192	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

757 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)(6,13)(7,18)(9,23)(10,11)(14,32)(15,22)(16,36)(17,31)(19,27)(20,29)(21,26)(25,28)(30,33)(34,35);(1,3,4,2)(5,6)(7,14)(8,22)(9,25)(10,16)(11,28)(12,17)(13,23)(15,26)(18,33)(19,31)(20,35)(21,34)(24,30)(27,36)(29,32);(5,7,12,21)(6,14,17,34)(8,23,24,19)(9,20,27,29)(10,26,11,18)(13,30,31,22)(15,28,33,16)(25,35,36,32);(5,8)(6,15)(7,19)(9,26)(10,20)(11,29)(12,24)(13,22)(14,16)(17,33)(18,27)(21,23)(25,35)(28,34)(30,31)(32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=deh,e^b=eh,h^b=hi,d^c=di,f^c=fi,e^d=eh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216402
maximal normal subgroup of	#72216402
maximal quotient of	#72226892

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405366);