#147307

order	512
nilpotency class	4
isoclinism	isoclinic class of #12952
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dhi,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=eh,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250645, #72250661, #72250663
maximal normal subgroup of	#72250645, #72250661, #72250663

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54503);