#147478

order	512
nilpotency class	4
isoclinism	isoclinic class of #13033
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

746 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)(7,12,17,27)(8,11)(9,22)(10,14)(13,19)(15,20)(16,23,31,34)(18,32,30,28)(21,24)(25,33)(26,36,35,29);(1,2,3,4)(6,11)(8,19)(9,20)(12,23)(13,15)(14,24)(16,29)(18,30)(21,33)(22,25)(26,28)(27,34)(31,36)(32,35);(5,7)(6,12,14,27)(8,16)(9,18)(10,17)(11,23,24,34)(13,26,25,35)(15,28,22,32)(19,29)(20,30)(21,31)(33,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,d^b=dh,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250249, #72250645, #72250663
maximal normal subgroup of	#72250249, #72250645, #72250663
maximal quotient of	#72307594

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54674);