#147409

order	512
nilpotency class	4
isoclinism	isoclinic class of #13077
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
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

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,4)(5,6)(7,18,21,39)(8,13)(9,28)(10,31)(11,17)(12,20,34,22)(14,29)(15,41)(16,32)(19,45,47,43)(23,53,50,44)(24,54)(25,57)(26,59)(27,38)(30,42)(33,46,61,49)(35,52,60,51)(36,56)(37,55)(40,58)(48,64,67,65)(62,68,63,66);(5,7,9,22,10,23,29,52,11,21,30,20,32,50,41,51)(6,12,15,18,16,35,42,44,17,34,14,39,31,60,28,53)(8,19,25,46,26,48,58,66,27,47,24,49,55,67,56,68)(13,33,36,43,37,62,54,64,38,61,40,45,59,63,57,65);(1,2,3,4)(5,8,11,27)(6,13,17,38)(7,19,21,47)(9,25,30,24)(10,26,32,55)(12,33,34,61)(14,40,15,36)(16,37,31,59)(18,43,39,45)(20,49,22,46)(23,48,50,67)(28,57,42,54)(29,58,41,56)(35,62,60,63)(44,64,53,65)(51,68,52,66)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54605);