#147698

order	512
nilpotency class	4
isoclinism	isoclinic class of #13402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250230, #72250231, #72250282, …
maximal normal subgroup of	#72250230, #72250231, #72250282, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54894);