#495851

order	512
nilpotency class	4
isoclinism	isoclinic class of #33688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

675 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,d^c=dh,f^c=fhi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224792, #72226756, #72226757, …
maximal normal subgroup of	#72224792, #72226756, #72226757, …
maximal quotient of	#72242479, #72242485

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403047);